# MATH APPLIED MATRIX THEORY

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points each:. Draw the web and its baclins. Solution: This is easy to do: just remember how a matrix describes a directed graph. For instance, since the element A(, 3 equals, then there is an arrow (baclin going from page to page 3. Since A(4,, then there is no baclin going from page 4 to page, and so on. 2. Find the importance score matrix B for the web. Solution: Once the web is understood, the importance scores x,..., x 5 verify the following x 3 x x 5 x 2 2 x + 3 x x 5 x 3 2 x In matrix notation, we obtain x x 2 x 3 x 4 x 5 x 4 x x 3 x 5 x 4 /3 /2 /2 /3 /2 /2 /3 x x 2 x 3 x 4 x 5

2 Hence, B /3 /2 /2 /3 /2 /2 /3 Notice that all the columns of B add up to (that is, B is column stochastic. This is a quic way to chec that B is not incorrect. 3. Let M.85 B +.5 /5 ones(5 and suppose that MATLAB gives >> [P D eig(m P i i i i i i i i D i i.482 Find the importance score vector x [x, x 2, x 3, x 4, x 5. Solution: The importance score vector must verify x Mx and x +x 2 +x 3 +x 4 +x 5. In particular, the equation x Mx says that x must be an eigenvector of M with eigenvalue λ. In the matrix D the eigenvalue λ is in the first column, so that one associated eigenvector will be the first column of P. That is, the vector v [.352,.499,.23,.5488,.538. Clearly, the sum of the components of this eigenvector v is not equal to. To create an eigenvector whose components add up to, we divide the vector v by the sum of its components. That is, x [.352,.499,.23,.5488, [.352,.499,.23,.5488, [.635,.233,.995,.2562,.2478 Hence the score importance vector is x [.635,.233,.995,.2562,

3 PROBLEM 2 (25 points Determine whether the following matrices are linearly independent M [ 2 (This is problem 2, section 7.2 [, M 2 [ 5 3, M [ 2, M 4 3 Solution: By definition, M, M 2, M 3, M 4 are linearly independent if whenever there are real numbers x, x 2, x 3, and x 4 such that x M + x 2 M 2 + x 3 M 3 + x 4 M 4, then necessarily x x 2 x 3 x 4. The equation x M + x 2 M 2 + x 3 M 3 + x 4 M 4 means ( x M + x 2 M 2 + x 3 M 3 + x 4 M 4. That is, x [ 2 + x 2 [ + x 3 [ x 4 [ 2 3 ( Thus, componentwise this implies That is, we end up with the system If we denote A x + x 2 + 5x 3 + 2x 4 x 2 + 3x 3 + x 4 x 2 3x 3 x 4 2x + x 2 + 5x 3 + 3x x x 2 x 3 x 4, then x is a solution to the system Ax. Linearly independence of M, M 2, M 3, and M 4 means that there is exactly one solution to Ax, that is, the zero solution. Let s see if that s the case. By computing the reduced row echelon form of the augmented matrix [A b, where b [,,,, (which is very simple to do by hand, we get that there is one free variable. Therefore, there are infinitely many solutions. This is telling us that M, M 2, M 3, and M 4 are linearly dependent since there are infinitely many non-zero numbers x, x 2, x 3, and x 4 such that x M + x 2 M 2 + x 3 M 3 + x 4 M 4. 3

4 PROBLEM 3 (3 points Determine whether the following sets are vector spaces ( points each. U {a + b t such that a, b R} (This is problem 28, section 7. Solution: By definition. U is a vector space if whenever the sum of two objects in U is also an object in U, and whenever an object is U is multiplied by a real number the result is also an object in U. To chec the first condition let s tae two objects u and v in U. By the definition of U u and v must be of the form u + l t, v r + s t where, l, r and s are some fixed real numbers (here t is the variable. Now, the sum u + v is u + v ( + r + (l + s t, which is also of the form real number + real number t. Hence, u + v is also an object of U. Now tae a real number λ and an object u U. Then, u must be of the form u m + y t for some real numbers m and y. Therefore, the product λu is given by λu λ(m + y t λm + λy t, which is also of the form real number + real number t. Hence, λu is also an object of U. Therefore, U is a vector space. {[ } a b 2. V such that a + d (This is problem 2, section 7. c d Solution: By definition, the objects of V are those 2 2 matrices whose diagonal elements add up to. Given two objects u and v in V, they must be of the form [ [ e f i j u, v g h l for some real numbers e, f, g, h, i, j, and l such that e + h and i + l. The sum u + v is then [ e + i f + j u + v g + h + l We only need to chec whether the diagonal elements of u + v add up to. The diagonal elements of u + v are e + i and h + l. Hence That, is u + v is also an object in V. (e + i + (h + l e + h + i + l +. Now, tae a real number λ and an object u V. Therefore, since u must be of the form [ p q u r s for some real numbers p, q, r, and s with p + s. The product λu is given by [ λp λq λu λr λs 4

5 It remains to chec whether the diagonal elements of λu add up to zero. We have λp + λs λ(p + s λ. Hence, λu is also an object of V, for every real number λ and every object u in V. Thus, V is a vector space. 3. The set S consisting of all 2 2 symmetric matrices. (This is problem 23, section 7. Solution: Given any two objects u and v of S, they must be of the form [ [ p q a b u, v q s b c for some real numbers p, q, s, a, b, and c. Notice that u and v are symmetric, that is, u u T and v v T (they equal their transposes. Now, u + v is given by [ p + a q + b u + v q + b s + c Notice that u + v is also symmetric. Therefore, u + v is also an object in S. Now tae a real number λ and an object u S. u must be of the form [ r t u t s for some real numbers r, s, and t. Hence, the product λu is given by [ λr λt λu, λt λs which is also a symmetric matrix. Hence, λu is also an object in S. Therefore, S is a vector space. 5

6 PROBLEM 4 (25 points The system of difference equations { x x 2y y 2x + y where, 2, 3,... defines a discrete dynamical system. Find a formula for u [x, y for every natural number using that x y and that MATLAB gives >> [P Deig([ -2; -2 (This is problem 2, section 6.3 [ / 2 / 2 P / 2 / 2 [ D 3 Solution: Set u [x, y. Clearly, the system above is equivalent to u Au, where the matrix A is given by A [ 2 2 Iterating the equation u Au we get that u A u (where u [x, y [,. On the other hand, by using the MATLAB information we see that A is diagonalizable (here we use the quic chec that its eigenvalues are all different. In particular, we have which gives [ A P D P P 3 A P D P, [ P ( P 3 P. [ a b Now remember that the inverse of any 2 2 matrix is given by the matrix c d ad bc [ d b c a Using this little tric, we easily compute the inverse of P. P ( / 2 (/ 2 ( / 2 ( / 2 [ / 2 / 2 / 2 / 2 [ / 2 / 2 / 2 / 2 as which gives that P P (that is, in this particular case P turned out to be an orthogonal matrix. All this gives [ A ( P 3 P. 6,

7 Hence, [ u A ( u P 3 Now, we compute the product [ / 2 / 2 / 2 / 2 [ / 2 / 2 P u / 2 / 2 [ ( 3 [ ( 3 [ ( / 2 / 2 / 2 / 2 [ / 2 / 2 [ [ / 2 / 2 / 2 / ( 2 3 / 2 + / 2 [ [ [ / 2 / 2 / 2 / ( 2/ [ [ / 2 / 2 / 2 / ( ( 2/ 2 2 [ ( / 2( ( 2/ 2 ( / 2( ( 2/ 2 [ ( ( [ / 2 / 2 / 2 / 2 [ ( Hence, the general formula for u is u (. That is y ( and x (. In particular, we have that x y, x 2 y 2, x 3 y 3, and so on. ( 7

8 [ PROBLEM 5 (2 points Determine whether the matrix P 2 matrix for A [ 4 2 is a diagonalizing. (This is problem 8 in the Review for Chapter 6, page 323 Solution: By definition, P is a diagonalizing matrix for A if the we have P A P D where D is a diagonal matrix. Hence, we only need to compute the product P A P and chec whether or not it is a diagonal matrix. [ a b To find P we use that the inverse of any 2 2 matrix is given by the matrix c d ad bc In our case, the matrix in question is P P [ d b c a [ 2 [ Hence, its inverse is given by [ 2 Now, we compute [ P A P 2 [ 2 [ 2 3 [ 4 2 [ [ 2 Hence, P A P is a diagonal matrix and therefore P is indeed a diagonalizing matrix for A. 8

9 PROBLEM 6 (2 points Suppose that the characteristic polynomial of a 3 3 matrix B is Answer the following (5 points each:. What are the eigenvalues of B? p(λ (λ (λ + (λ 4. Solution: By definition, the eigenvalues of B are the roots of its characteristic polynomial. That is, the eigenvalue of B are the solutions to Clearly, the eigenvalues are,, and Is B diagonalizable? p(λ (λ (λ + (λ 4 Solution: Yes, it is. The eigenvalues are all distinct (they do not repeat. This is a sufficient (although not necessary condition for diagonalization. 3. What is det(b? Solution: Since B is diagonalizable, we now that B P D P for some invertible matrix P and some diagonal matrix D whose diagonal elements are the eigenvalues of A. Therefore we have det(b det(p D P det(p det(d det(p det(p det(d det(p Thus, det(b Is B invertible? det(d ( 4 4. Solution: Yes, since det(b 4 it follows that B is invertible. 9

10 PROBLEM 7 (25 points Let u [ 3, 2, 4, v [ 2,, and w [, 5,. Compute u (v w. (This is problem 56, section 5.2 Solution: Let s recall the definitions of the dot and cross products (which is a number and [x, x 2, x 3 [y, y 2, y 3 x y + x 2 y 2 + x 3 y 3, [x, x 2, x 3 [y, y 2, y 3 [x 2 y 3 x 3 y 2, x 3 y x y 3, x y 2 x 2 y (which is a vector. We then have v w [ 2,, [, 5, [4, 2, and u (v w [ 3, 2, 4 [4, 2, PROBLEM 8 (25 points Find the shortest distance from the point P ( 2, 3, to the subspace U span{u, u 2 } where u [2,, and u 2 [7, 4, 2. (This is problem 7, section 4.3 Solution: Let v [ 2, 3,. Since u and u 2 are orthogonal (their dot product is zero the projection of v onto the subspace U is given by v v u u 2 u + v u 2 u 2 2 u 2 Recall that the norm of a vector x [x, x 2, x 3 is given by x x 2 + x2 2 + x2 3. Computing and plugging in the dot products and the norms in the formula for v we obtain v 7 5 u u [2,, [7, 4, 2. Finally, the distance from P to the subspace U is given by v v

11 PROBLEM 9 (3 points Let T : R 3 R 2 be the linear operator given by T (x, x 2, x 3 (2x 3x 2 + x 3, x + x 3. Determine the following ( points each (This is problem 7, section 3.4:. The matrix that represents T. Solution: The canonical basis for R 3 is B {[,,, [,,, [,, } Hence, the matrix A that represents T is given by A [T (,,, T (,,, T (,, We now use the definition of T to compute ( T (,, + and Hence, 2. What is ran(t? Is T onto? ( T (,, + ( T (,, + A ( 2 3,,,. Solution: ran(t equals col(a. Hence, since a basis for col(a is {( ( } 2 3, (the third column is a linear combination of the first two. We obtain that {( ( } 2 3 col(a span, In particular, ran(a 2, which coincides with the dimension of the target space R 2. Therefore, T is onto. 3. What is er(t? Is T one-to-one? Solution: Since er(t null(a and, by definition, null(a consists of those vectors x such that Ax, we obtain that er(t is the subspace formed by the solutions to Ax. This leads to the system ( 2 3 x x 2 x 3 (.

12 Computing the reduced row echelon form of the augmented matrix ( [A [, 2 3, which can be easily done by hand, we obtain that there is one free variable and that er(t turns out to be er(t span ( [3, 3. By definition, T is one-to-one only when er(t consists of just the zero vector. In our case er(t consists of infinitely many vectors (all the multiples of [3, 3. Hence, T is not one-to-one. PROBLEM (2 points Find the value of such that the angle between the vectors [2, 3, and [ 2, 5, equals π/2. (This is problem 7, section 4. Solution: The vectors forming an angle of π/2 means that they are orthogonal to each other. That is, their dot product has to be zero. Hence, we need to find the value of such that Clearly the answer is 9. [2, 3, [ 2, 5,

### 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

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### 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

### 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,

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### 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

### 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

### Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### 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

### MAT188H1S Lec0101 Burbulla

Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### 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

### Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

### Lecture notes on linear algebra

Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra

### x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r

### 3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

### 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

### Manifold Learning Examples PCA, LLE and ISOMAP

Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition

### Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products

Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products H. Geuvers Institute for Computing and Information Sciences Intelligent Systems Version: spring 2015 H. Geuvers Version:

### LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### APPLICATIONS. are symmetric, but. are not.

CHAPTER III APPLICATIONS Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes In symbols, A t =

### 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

### Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

### 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

### is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### 6. Cholesky factorization

6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Mathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.

Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### 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

### Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### 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

### 26. Determinants I. 1. Prehistory

26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

### Multivariate Analysis of Variance (MANOVA): I. Theory

Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the

### Orthogonal Projections and Orthonormal Bases

CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### 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

### Solutions to Assignment 10

Soltions to Assignment Math 27, Fall 22.4.8 Define T : R R by T (x) = Ax where A is a matrix with eigenvales and -2. Does there exist a basis B for R sch that the B-matrix for T is a diagonal matrix? We

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

### Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

### Answers to exercises LINEAR ALGEBRA Jim Hefferon

Answers to exercises LINEAR ALGEBRA Jim Hefferon http://joshua.smcvt.edu/linearalgebra Notation R, R +, R n real numbers, positive reals, n-tuples of reals N, C natural numbers {,, 2,...}, complex numbers

### Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### A tutorial on Principal Components Analysis

A tutorial on Principal Components Analysis Lindsay I Smith February 26, 2002 Chapter 1 Introduction This tutorial is designed to give the reader an understanding of Principal Components Analysis (PCA).

### SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIX-EXPONENTIAL DISTRIBUTIONS

Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 1532-6349 print/1532-4214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL

### Vector Spaces. Chapter 2. 2.1 R 2 through R n

Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final

### Subspaces of R n LECTURE 7. 1. Subspaces

LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r

### Orthogonal Bases and the QR Algorithm

Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook

Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a

### Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

### Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

### CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.

CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### The Projection Matrix

The Projection Matrix David Arnold Fall 996 Abstract In this activity you will use Matlab to project a set of vectors onto a single vector. Prerequisites. Inner product (dot product) and orthogonal vectors.

### We shall turn our attention to solving linear systems of equations. Ax = b

59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

### 4.3 Least Squares Approximations

18 Chapter. Orthogonality.3 Least Squares Approximations It often happens that Ax D b has no solution. The usual reason is: too many equations. The matrix has more rows than columns. There are more equations

### Pythagorean vectors and their companions. Lattice Cubes

Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Structure of the Root Spaces for Simple Lie Algebras

Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### (67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT