# The design matrix X is a rectangular matrix of order m by n with elements. x i,j = ϕ j (t i ).

Save this PDF as:

Size: px
Start display at page:

Download "The design matrix X is a rectangular matrix of order m by n with elements. x i,j = ϕ j (t i )."

## Transcription

1 Chapter 5 Least Squares The term least squares describes a frequently used approach to solving overdetermined or inexactly specified systems of equations in an approximate sense. Instead of solving the equations exactly, we seek only to minimize the sum of the squares of the residuals. The least squares criterion has important statistical interpretations. If appropriate probabilistic assumptions about underlying error distributions are made, least squares produces what is known as the maximum-likelihood estimate of the parameters. Even if the probabilistic assumptions are not satisfied, years of experience have shown that least squares produces useful results. The computational techniques for linear least squares problems make use of orthogonal matrix factorizations. 5.1 Models and Curve Fitting A very common source of least squares problems is curve fitting. Let t be the independent variable and let y(t) denote an unknown function of t that we want to approximate. Assume there are m observations, i.e., values of y measured at specified values of t: y i = y(t i ), i = 1,..., m. The idea is to model y(t) by a linear combination of n basis functions: y(t) β 1 ϕ 1 (t) + + β n ϕ n (t). The design matrix X is a rectangular matrix of order m by n with elements x i,j = ϕ j (t i ). The design matrix usually has more rows than columns. In matrix-vector notation, the model is y Xβ. September 17,

2 2 Chapter 5. Least Squares The symbol stands for is approximately equal to. We are more precise about this in the next section, but our emphasis is on least squares approximation. The basis functions ϕ j (t) can be nonlinear functions of t, but the unknown parameters, β j, appear in the model linearly. The system of linear equations Xβ y is overdetermined if there are more equations than unknowns. The Matlab backslash operator computes a least squares solution to such a system. beta = X\y The basis functions might also involve some nonlinear parameters, α 1,..., α p. The problem is separable if it involves both linear and nonlinear parameters: y(t) β 1 ϕ 1 (t, α) + + β n ϕ n (t, α). The elements of the design matrix depend upon both t and α: x i,j = ϕ j (t i, α). Separable problems can be solved by combining backslash with the Matlab function fminsearch or one of the nonlinear minimizers available in the Optimization Toolbox. The new Curve Fitting Toolbox provides a graphical interface for solving nonlinear fitting problems. Some common models include the following: Straight line: If the model is also linear in t, it is a straight line: y(t) β 1 t + β 2. Polynomials: The coefficients β j appear linearly. Matlab orders polynomials with the highest power first: ϕ j (t) = t n j, j = 1,..., n, y(t) β 1 t n β n 1 t + β n. The Matlab function polyfit computes least squares polynomial fits by setting up the design matrix and using backslash to find the coefficients. Rational functions: The coefficients in the numerator appear linearly; the coefficients in the denominator appear nonlinearly: ϕ j (t) = t n j α 1 t n α n 1 t + α n, y(t) β 1t n β n 1 t + β n α 1 t n α n 1 t + α n.

3 5.2. Norms 3 Exponentials: The decay rates, λ j, appear nonlinearly: ϕ j (t) = e λjt, y(t) β 1 e λ 1t + + β n e λ nt. Log-linear: If there is only one exponential, taking logs makes the model linear but changes the fit criterion: y(t) Ke λt, log y β 1 t + β 2, with β 1 = λ, β 2 = log K. Gaussians: The means and variances appear nonlinearly: ϕ j (t) = e ( t µ j σ j ) 2, ( y(t) β 1 e t µ1 σ 1 ) 2 t µn + + β n e ( σn ) Norms The residuals are the differences between the observations and the model: r i = y i Or, in matrix-vector notation, n β j ϕ j (t i, α), i = 1,..., m. 1 r = y X(α)β. We want to find the α s and β s that make the residuals as small as possible. What do we mean by small? In other words, what do we mean when we use the symbol? There are several possibilities. Interpolation: If the number of parameters is equal to the number of observations, we might be able to make the residuals zero. For linear problems, this will mean that m = n and that the design matrix X is square. If X is nonsingular, the β s are the solution to a square system of linear equations: β = X \y. Least squares: Minimize the sum of the squares of the residuals: r 2 = m ri 2. 1

4 4 Chapter 5. Least Squares Weighted least squares: If some observations are more important or more accurate than others, then we might associate different weights, w j, with different observations and minimize r 2 w = m w i ri 2. 1 For example, if the error in the ith observation is approximately e i, then choose w i = 1/e i. Any algorithm for solving an unweighted least squares problem can be used to solve a weighted problem by scaling the observations and design matrix. We simply multiply both y i and the ith row of X by w i. In Matlab, this can be accomplished with X = diag(w)*x y = diag(w)*y One-norm: Minimize the sum of the absolute values of the residuals: r 1 = m r i. 1 This problem can be reformulated as a linear programming problem, but it is computationally more difficult than least squares. The resulting parameters are less sensitive to the presence of spurious data points or outliers. Infinity-norm: Minimize the largest residual: r = max r i. i This is also known as a Chebyshev fit and can be reformulated as a linear programming problem. Chebyshev fits are frequently used in the design of digital filters and in the development of approximations for use in mathematical function libraries. The Matlab Optimization and Curve Fitting Toolboxes include functions for one-norm and infinity-norm problems. We will limit ourselves to least squares in this book. 5.3 censusgui The NCM program censusgui involves several different linear models. The data are the total population of the United States, as determined by the U.S. Census, for the years 1900 to The units are millions of people.

5 5.3. censusgui 5 t y The task is to model the population growth and predict the population when t = The default model in censusgui is a cubic polynomial in t: y β 1 t 3 + β 2 t 2 + β 3 t + β 4. There are four unknown coefficients, appearing linearly. 400 Predict U.S. Population in Millions Figure 5.1. censusgui. Numerically, it s a bad idea to use powers of t as basis functions when t is around 1900 or The design matrix is badly scaled and its columns are nearly linearly dependent. A much better basis is provided by powers of a translated and scaled t: s = (t 1955)/55.

6 6 Chapter 5. Least Squares This new variable is in the interval 1 s 1 and the model is y β 1 s 3 + β 2 s 2 + β 3 s + β 4. The resulting design matrix is well conditioned. Figure 5.1 shows the fit to the census data by the default cubic polynomial. The extrapolation to the year 2020 seems reasonable. The push buttons allow you to vary the degree of the polynomial. As the degree increases, the fit becomes more accurate in the sense that r decreases, but it also becomes less useful because the variation between and outside the observations increases. The censusgui menu also allows you to choose interpolation by spline and pchip and to see the log-linear fit y Ke λt. Nothing in the censusgui tool attempts to answer the all-important question, Which is the best model? That s up to you to decide. 5.4 Householder Reflections Householder reflections are matrix transformations that are the basis for some of the most powerful and flexible numerical algorithms known. We will use Householder reflections in this chapter for the solution of linear least squares problems and in a later chapter for the solution of matrix eigenvalue and singular value problems. Formally, a Householder reflection is a matrix of the form H = I ρuu T, where u is any nonzero vector and ρ = 2/ u 2. The quantity uu T is a matrix of rank one where every column is a multiple of u and every row is a multiple of u T. The resulting matrix H is both symmetric and orthogonal, that is, and H T = H H T H = H 2 = I. In practice, the matrix H is never formed. Instead, the application of H to a vector x is computed by τ = ρu T x, Hx = x τu. Geometrically, the vector x is projected onto u and then twice that projection is subtracted from x. Figure 5.2 shows a vector u and a line labeled u that is perpendicular to u. It also shows two vectors, x and y, and their images, Hx and Hy, under the

7 5.4. Householder Reflections 7 Householder reflection u x u Hx y Hy Figure 5.2. Householder reflection. transformation H. The matrix transforms any vector into its mirror image in the line u. For any vector x, the point halfway between x and Hx, that is, the vector x (τ/2)u, is actually on the line u. In more than two dimensions, u is the plane perpendicular to the defining vector u. Figure 5.2 also shows what happens if u bisects the angle between x and one of the axes. The resulting Hx then falls on that axis. In other words, all but one of the components of Hx are zero. Moreover, since H is orthogonal, it preserves length. Consequently, the nonzero component of Hx is ± x. For a given vector x, the Householder reflection that zeros all but the kth component of x is given by σ = ± x, u = x + σe k, ρ = 2/ u 2 = 1/(σu k ), H = I ρuu T. In the absence of roundoff error, either sign could be chosen for σ, and the resulting Hx would be on either the positive or the negative kth axis. In the presence of roundoff error, it is best to choose the sign so that sign σ = sign x k. Then the operation x k + σ is actually an addition, not a subtraction.

8 8 Chapter 5. Least Squares 5.5 The QR Factorization If all the parameters appear linearly and there are more observations than basis functions, we have a linear least squares problem. The design matrix X is m by n with m > n. We want to solve Xβ y. But this system is overdetermined there are more equations than unknowns. So we cannot expect to solve the system exactly. Instead, we solve it in the least squares sense: min β Xβ y. A theoretical approach to solving the overdetermined system begins by multiplying both sides by X T. This reduces the system to a square, n-by-n system known as the normal equations: X T Xβ = X T y. If there are thousands of observations and only a few parameters, the design matrix X is quite large, but the matrix X T X is small. We have projected y onto the space spanned by the columns of X. Continuing with this theoretical approach, if the basis functions are independent, then X T X is nonsingular and β = (X T X ) 1 X T y. This formula for solving linear least squares problems appears in most textbooks on statistics and numerical methods. However, there are several undesirable aspects to this theoretical approach. We have already seen that using a matrix inverse to solve a system of equations is more work and less accurate than solving the system by Gaussian elimination. But, more importantly, the normal equations are always more badly conditioned than the original overdetermined system. In fact, the condition number is squared: κ(x T X ) = κ(x ) 2. With finite-precision computation, the normal equations can actually become singular, and (X T X ) 1 nonexistent, even though the columns of X are independent. As an extreme example, consider the design matrix X = 1 1 δ 0. 0 δ If δ is small, but nonzero, the two columns of X are nearly parallel but are still linearly independent. The normal equations make the situation worse: ( ) X T 1 + δ 2 1 X = δ 2.

9 5.5. The QR Factorization 9 If δ < 10 8, the matrix X T X computed with double-precision floating-point arithmetic is exactly singular and the inverse required in the classic textbook formula does not exist. Matlab avoids the normal equations. The backslash operator not only solves square, nonsingular systems, but also computes the least squares solution to rectangular, overdetermined systems: β = X \y. Most of the computation is done by an orthogonalization algorithm known as the QR factorization. The factorization is computed by the built-in function qr. The NCM function qrsteps demonstrates the individual steps. The two versions of the QR factorization are illustrated in Figure 5.3. Both versions have X = QR. In the full version, R is the same size as X and Q is a square matrix with as many rows as X. In the economy-sized version, Q is the same size as X and R is a square matrix with as many columns as X. The letter Q is a substitute for the letter O in orthogonal and the letter R is for right triangular matrix. The Gram Schmidt process described in many linear algebra texts is a related, but numerically less satisfactory, algorithm that generates the same factorization. X = Q R X = Q R Figure 5.3. Full and economy QR factorizations. A sequence of Householder reflections is applied to the columns of X to produce the matrix R: H n H 2 H 1 X = R.

10 10 Chapter 5. Least Squares The jth column of R is a linear combination of the first j columns of X. Consequently, the elements of R below the main diagonal are zero. If the same sequence of reflections is applied to the right-hand side, the equations Xβ y become where Rβ z, H n H 2 H 1 y = z. The first n of these equations is a small, square, triangular system that can be solved for β by back substitution with the subfunction backsubs in bslashtx. The coefficients in the remaining m n equations are all zero, so these equations are independent of β and the corresponding components of z constitute the transformed residual. This approach is preferable to the normal equations because Householder reflections have impeccable numerical credentials and because the resulting triangular system is ready for back substitution. The matrix Q in the QR factorization is Q = (H n H 2 H 1 ) T. To solve least squares problems, we do not have to actually compute Q. In other uses of the factorization, it may be convenient to have Q explicitly. If we compute just the first n columns, we have the economy-sized factorization. If we compute all m columns, we have the full factorization. In either case, Q T Q = I, so Q has columns that are perpendicular to each other and have unit length. Such a matrix is said to have orthonormal columns. For the full Q, it is also true that QQ T = I, so the full Q is an orthogonal matrix. Let s illustrate this with a small version of the census example. We will fit the six observations from 1950 to 2000 with a quadratic: y(s) β 1 s 2 + β 2 s + β 3. The scaled time s = ((1950:10:2000) )/50 and the observations y are s y

11 5.5. The QR Factorization 11 The design matrix is X = [s.*s s ones(size(s))] The M-file qrsteps shows the steps in the QR factorization. qrsteps(x,y) The first step introduces zeros below the diagonal in the first column of X The same Householder reflection is applied to y Zeros are introduced in the second column The second Householder reflection is also applied to y Finally, zeros are introduced in the third column and the reflection applied to y. This produces the triangular matrix R and a modified right-hand side z.

12 12 Chapter 5. Least Squares R = z = The system of equations Rβ = z is the same size as the original, 6 by 3. We can solve the first three equations exactly (because R(1 : 3, 1 : 3) is nonsingular). beta = R(1:3,1:3)\z(1:3) beta = This is the same solution beta that the backslash operator computes with or beta = R\z beta = X\y The last three equations in Rβ = z cannot be satisfied by any choice of β, so the last three components of z represent the residual. In fact, the two quantities norm(z(4:6)) norm(x*beta - y) are both equal to Notice that even though we used the QR factorization, we never actually computed Q. The population in the year 2010 can be predicted by evaluating β 1 s 2 + β 2 s + β 3 at s = ( )/50 = 1.2. This can be done with polyval. p2010 = polyval(beta,1.2) p2010 = The actual 2010 census found the population to be million.

13 5.6. Pseudoinverse Pseudoinverse The definition of the pseudoinverse makes use of the Frobenius norm of a matrix: ( 1/2 A F = ai,j) 2. i j The Matlab expression norm(x, fro ) computes the Frobenius norm. A F the same as the 2-norm of the long vector formed from all the elements of A. is norm(a, fro ) == norm(a(:)) The Moore Penrose pseudoinverse generalizes and extends the usual matrix inverse. The pseudoinverse is denoted by a dagger superscript, and computed by the Matlab pinv. Z = pinv(x) Z = X, If X is square and nonsingular, then the pseudoinverse and the inverse are the same: X = X 1. If X is m by n with m > n and X has full rank, then its pseudoinverse is the matrix involved in the normal equations: X = (X T X ) 1 X T. The pseudoinverse has some, but not all, of the properties of the ordinary inverse. X is a left inverse because X X = (X T X ) 1 X T X = I is the n-by-n identity. But X is not a right inverse because XX = X(X T X ) 1 X T only has rank n and so cannot be the m-by-m identity. The pseudoinverse does get as close to a right inverse as possible in the sense that, out of all the matrices Z that minimize Z = X also minimizes XZ I F, Z F. It turns out that these minimization properties also define a unique pseudoinverse even if X is rank deficient.

14 14 Chapter 5. Least Squares Consider the 1-by-1 case. What is the inverse of a real (or complex) number x? If x is not zero, then clearly x 1 = 1/x. But if x is zero, x 1 does not exist. The pseudoinverse takes care of that because, in the scalar case, the unique number that minimizes both xz 1 and z is x = { 1/x : x 0, 0 : x = 0. The actual computation of the pseudoinverse involves the singular value decomposition, which is described in a later chapter. You can edit pinv or type pinv to see the code. 5.7 Rank Deficiency If X is rank deficient, or has more columns than rows, the square matrix X T X is singular and (X T X ) 1 does not exist. The formula β = (X T X ) 1 X T y obtained from the normal equations breaks down completely. In these degenerate situations, the least squares solution to the linear system Xβ y is not unique. A null vector of X is a nonzero solution to Xη = 0. Any multiple of any null vector can be added to β without changing how well Xβ approximates y. In Matlab, the solution to Xβ y can be computed with either backslash or the pseudoinverse, that is, or beta = X\y beta = pinv(x)*y In the full rank case, these two solutions are the same, although pinv does considerably more computation to obtain it. But in degenerate situations these two solutions are not the same. The solution computed by backslash is called a basic solution. If r is the rank of X, then at most r of the components of beta = X\y

15 5.7. Rank Deficiency 15 are nonzero. Even the requirement of a basic solution does not guarantee uniqueness. The particular basic computation obtained with backslash is determined by details of the QR factorization. The solution computed by pinv is the minimum norm solution. Out of all the vectors β that minimize Xβ y, the vector beta = pinv(x)*y also minimizes β. This minimum norm solution is unique. For example, let X = and y = The matrix X is rank deficient. The middle column is the average of the first and last columns. The vector is a null vector. The statement beta = X\y produces a warning, η = Warning: Rank deficient, rank = 2 tol = e-14 and the solution beta = As promised, this solution is basic; it has only two nonzero components. However, the vectors beta =

16 16 Chapter 5. Least Squares and beta = are also basic solutions. The statement beta = pinv(x)*y produces the solution beta = without giving a warning about rank deficiency. The norm of the pseudoinverse solution norm(pinv(x)*y) = is slightly less than the norm of the backslash solution norm(x\y) = Out of all the vectors β that minimize Xβ y, the pseudoinverse has found the shortest. Notice that the difference between the two solutions, X\y - pinv(x)*y = is a multiple of the null vector η. If handled with care, rank deficient least squares problems can be solved in a satisfactory manner. Problems that are nearly, but not exactly, rank deficient are more difficult to handle. The situation is similar to square linear systems that are badly conditioned, but not exactly singular. Such problems are not well posed numerically. Small changes in the data can lead to large changes in the computed solution. The algorithms used by both backslash and pseudoinverse involve decisions about linear independence and rank. These decisions use somewhat arbitrary tolerances and are particularly susceptible to both errors in the data and roundoff errors in the computation. Which is better, backslash or pseudoinverse? In some situations, the underlying criteria of basic solution or minimum norm solution may be relevant. But most problem formulations, particularly those involving curve fitting, do not include such subtle distinctions. The important fact to remember is that the computed solutions are not unique and are not well determined by the data.

17 5.8. Separable Least Squares Separable Least Squares Matlab provides several functions for solving nonlinear least squares problems. Older versions of Matlab have one general-purpose, multidimensional, nonlinear minimizer, fmins. In more recent versions of Matlab, fmins has been updated and renamed fminsearch. The Optimization Toolbox provides additional capabilities, including a minimizer for problems with constraints, fmincon; a minimizer for unconstrained problems, fminunc; and two functions intended specifically for nonlinear least squares, lsqnonlin and lsqcurvefit. The Curve Fitting Toolbox provides a GUI to facilitate the solution of many different linear and nonlinear fitting problems. In this introduction, we focus on the use of fminsearch. This function uses a direct search technique known as the Nelder Meade algorithm. It does not attempt to approximate any gradients or other partial derivatives. It is quite effective on small problems involving only a few variables. Larger problems with more variables are better handled by the functions in the Optimization and Curve Fitting Toolboxes. Separable least squares curve-fitting problems involve both linear and nonlinear parameters. We could ignore the linear portion and use fminsearch to search for all the parameters. But if we take advantage of the separable structure, we obtain a more efficient, robust technique. With this approach, fminsearch is used to search for values of the nonlinear parameters that minimize the norm of the residual. At each step of the search process, the backslash operator is used to compute values of the linear parameters. Two blocks of Matlab code are required. One block can be a function, a script, or a few lines typed directly in the Command Window. It sets up the problem, establishes starting values for the nonlinear parameters, calls fminsearch, processes the results, and usually produces a plot. The second block is the objective function that is called by fminsearch. This function is given a vector of values of the nonlinear parameters, alpha. It should compute the design matrix X for these parameters, use backslash with X and the observations to compute values of the linear parameters beta, and return the resulting residual norm. Let s illustrate all this with expfitdemo, which involves observations of radioactive decay. The task is to model the decay by a sum of two exponential terms with unknown rates λ j : y β 1 e λ1t + β 2 e λ2t. Consequently, in this example, there are two linear parameters and two nonlinear parameters. The demo plots the various fits that are generated during the nonlinear minimization process. Figure 5.4 shows plots of both the data and the final fit. The outer function begins by specifying 21 observations, t and y. function expfitdemo t = (0:.1:2) ; y = [ ] ;

18 18 Chapter 5. Least Squares Figure 5.4. expfitdemo. The initial plot uses o s for the observations, creates an all-zero placeholder for what is going to become the evolving fit, and creates a title that will show the values of lambda. The variable h holds the handles for these three graphics objects. clf shg set(gcf, doublebuffer, on ) h = plot(t,y, o,t,0*t, - ); h(3) = title( ); axis([ ]) The vector lambda0 specifies initial values for the nonlinear parameters. In this example, almost any choice of initial values leads to convergence, but in other situations, particularly with more nonlinear parameters, the choice of initial values can be much more important. The call to fminsearch does most of the work. The observations t and y, as well as the graphics handle h, remain fixed throughout the calculation and are accessible by the inner function expfitfun. lambda0 = [3 6] ; lambda = set(h(2), color, black ) The inner objective function is named expfitfun. It can handle n exponential basis functions; we will be using n = 2. The single input parameter is a vector

19 5.9. Further Reading 19 provided by fminsearch that contains values of the n decay rates, λ j. The function computes the design matrix, uses backslash to compute β, evaluates the resulting model, and returns the norm of the residual. function res = expfitfun(lambda) m = length(t); n = length(lambda); X = zeros(m,n); for j = 1:n X(:,j) = exp(-lambda(j)*t); end beta = X\y; z = X*beta; res = norm(z-y); The objective function also updates the plot of the fit and the title and pauses long enough for us to see the progress of the computation. set(h(2), ydata,z); set(h(3), string,sprintf( %8.4f %8.4f,lambda)) pause(.1) end 5.9 Further Reading The reference books on matrix computation [4, 5, 6, 7, 8, 9] discuss least squares. An additional reference is Björck [1]. Exercises 5.1. Let X be the n-by-n matrix generated by [I,J] = ndgrid(1:n); X = min(i,j) + 2*eye(n,n) - 2; (a) How does the condition number of X grow with n? (b) Which, if any, of the triangular factorizations chol(x), lu(x), and qr(x) reveal the poor conditioning? 5.2. In censusgui, change the 1950 population from million to million. This produces an extreme outlier in the data. Which models are the most affected by this outlier? Which models are the least affected? 5.3. If censusgui is used to fit the U.S. Census data with a polynomial of degree eight and the fit is extrapolated beyond the year 2000, the predicted population actually becomes zero before the year On what year, month, and day does that fateful event occur?

20 20 Chapter 5. Least Squares 5.4. Here are some details that we skipped over in our discussion of Householder reflections. At the same time, we extend the description to include complex matrices. The notation u T for transpose is replaced with the Matlab notation u for complex conjugate transpose. Let x be any nonzero m-by-1 vector and let e k denote the kth unit vector, that is, the kth column of the m-by-m identity matrix. The sign of a complex number z = re iθ is Define σ by Let 5.5. Let sign(z) = z/ z = e iθ. σ = sign(x k ) x. u = x + σe k. In other words, u is obtained from x by adding σ to its kth component. (a) The definition of ρ uses σ, the complex conjugate of σ: Show that ρ = 1/( σu k ). ρ = 2/ u 2. (b) The Householder reflection generated by the vector x is Show that and that H = I ρuu. H = H H H = I. (c) Show that all the components of Hx are zero, except for the kth. In other words, show that Hx = σe k. (d) For any vector y, let Show that τ = ρu y. Hy = y τu. x = (a) Find the Householder reflection H that transforms x into Hx =

21 Exercises Let (b) Find nonzero vectors u and v that satisfy X = Hu = u, Hv = v. (a) Verify that X is rank deficient Consider three choices for the pseudoinverse of X. Z = pinv(x) B = X\eye(5,5) S = eye(3,3)/x (b) Compare the values of % The actual pseudoinverse % Backslash % Slash Z F, B F, and S F ; XZ I F, XB I F, and XS I F ; ZX I F, BX I F, and SX I F. Verify that the values obtained with Z are less than or equal to the values obtained with the other two choices. Actually minimizing these quantities is one way of characterizing the pseudoinverse. (c) Verify that Z satisfies all four of the following conditions, and that B and S fail to satisfy at least one of the conditions. These conditions are known as the Moore Penrose equations and are another way to characterize a unique pseudoinverse. XZ is symmetric. ZX is symmetric. XZX = X. ZXZ = Z Generate 11 data points, t k = (k 1)/10, y k = erf(t k ), k = 1,..., 11. (a) Fit the data in a least squares sense with polynomials of degrees 1 through 10. Compare the fitted polynomial with erf(t) for values of t between the data points. How does the maximum error depend on the polynomial degree? (b) Because erf(t) is an odd function of t, that is, erf(x) = erf( x), it is reasonable to fit the data by a linear combination of odd powers of t: erf(t) c 1 t + c 2 t c n t 2n 1. Again, see how the error between data points depends on n.

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction

### CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### 3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

### Using the Singular Value Decomposition

Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces

### Syntax. Description. MATLAB Function Reference Arithmetic Operators + - * / \ ^ ' Matrix and array arithmetic

MATLAB Function Reference Arithmetic Operators + - * / \ ^ ' Matrix and array arithmetic Syntax A+B A-B A*B A.*B A/B A./B A\B A.\B A^B A.^B A' A.' Description MATLAB has two different types of arithmetic

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### Eigenvalues and Eigenvectors

Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution

### Solution of Linear Systems

Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

### Linear Least Squares

Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Nonlinear Iterative Partial Least Squares Method

Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

### Solving Sets of Equations. 150 B.C.E., 九章算術 Carl Friedrich Gauss,

Solving Sets of Equations 5 B.C.E., 九章算術 Carl Friedrich Gauss, 777-855 Gaussian-Jordan Elimination In Gauss-Jordan elimination, matrix is reduced to diagonal rather than triangular form Row combinations

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every

### Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

### The Image Deblurring Problem

page 1 Chapter 1 The Image Deblurring Problem You cannot depend on your eyes when your imagination is out of focus. Mark Twain When we use a camera, we want the recorded image to be a faithful representation

### Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

### Solving Linear Systems of Equations. Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.

Solving Linear Systems of Equations Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.edu These slides are a supplement to the book Numerical Methods with Matlab:

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Common Curriculum Map. Discipline: Math Course: College Algebra

Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a

### CD-ROM Appendix E: Matlab

CD-ROM Appendix E: Matlab Susan A. Fugett Matlab version 7 or 6.5 is a very powerful tool useful for many kinds of mathematical tasks. For the purposes of this text, however, Matlab 7 or 6.5 will be used

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### NUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS

NUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS DO Q LEE Abstract. Computing the solution to Least Squares Problems is of great importance in a wide range of fields ranging from numerical

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Linear Algebra Review. Vectors

Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Canonical Correlation

Chapter 400 Introduction Canonical correlation analysis is the study of the linear relations between two sets of variables. It is the multivariate extension of correlation analysis. Although we will present

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### Mean value theorem, Taylors Theorem, Maxima and Minima.

MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### pp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64

Semester 1 Text: Chapter 1: Tools of Algebra Lesson 1-1: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 1-2: Algebraic Expressions

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Advanced Higher Mathematics Course Assessment Specification (C747 77)

Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

### A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### Lecture 5: Singular Value Decomposition SVD (1)

EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

### Numerical Linear Algebra Chap. 4: Perturbation and Regularisation

Numerical Linear Algebra Chap. 4: Perturbation and Regularisation Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation TUHH Heinrich Voss Numerical Linear

### Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

### Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Contents. The Real Numbers. Linear Equations and Inequalities in One Variable

dug33513_fm.qxd 11/20/07 3:21 PM Page vii Preface Guided Tour: Features and Supplements Applications Index 1 2 The Real Numbers 1.1 1.2 1.3 1.4 1.5 1.6 1 Sets 2 The Real Numbers 9 Operations on the Set

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### 1 Review of Least Squares Solutions to Overdetermined Systems

cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

### Least-Squares Intersection of Lines

Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a

### = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

### Practical Numerical Training UKNum

Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

### Linear Systems COS 323

Linear Systems COS 323 Last time: Constrained Optimization Linear constrained optimization Linear programming (LP) Simplex method for LP General optimization With equality constraints: Lagrange multipliers

### Identify examples of field properties: commutative, associative, identity, inverse, and distributive.

Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### The world s largest matrix computation. (This chapter is out of date and needs a major overhaul.)

Chapter 7 Google PageRank The world s largest matrix computation. (This chapter is out of date and needs a major overhaul.) One of the reasons why Google TM is such an effective search engine is the PageRank

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number

### u = [ 2 4 5] has one row with three components (a 3 v = [2 4 5] has three rows separated by semicolons (a 3 w = 2:5 generates the row vector w = [ 2 3

MATLAB Tutorial You need a small numb e r of basic commands to start using MATLAB. This short tutorial describes those fundamental commands. You need to create vectors and matrices, to change them, and

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### Operation Count; Numerical Linear Algebra

10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point

### Factor Analysis. Chapter 420. Introduction

Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### The Second Undergraduate Level Course in Linear Algebra

The Second Undergraduate Level Course in Linear Algebra University of Massachusetts Dartmouth Joint Mathematics Meetings New Orleans, January 7, 11 Outline of Talk Linear Algebra Curriculum Study Group

### Algebrator Manual Softmath

Algebrator Manual I Algebrator Manual Table of Contents Foreword 0 Part I Quick introduction to the Algebrator software 1 1 What... is new in version 5.0 (NEW features) 2 2 Getting... technical support

### ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### Applied Linear Algebra I Review page 1

Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

### EC327: Advanced Econometrics, Spring 2007

EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Appendix D: Summary of matrix algebra Basic definitions A matrix is a rectangular array of numbers, with m

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

### Matrix Differentiation

1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides