# Math 215 HW #7 Solutions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Math 5 HW #7 Solutions Problem 8 If P is the projection matrix onto a k-dimensional subspace S of the whole space R n, what is the column space of P and what is its rank? Answer: The column space of P is S To see this, notice that, if x R n, then P x S since P projects x to S Therefore, col(p ) S On the other hand, if b S, then P b b, so S col(p ) Since containment goes both ways, we see that col(p ) S Therefore, since the rank of P is equal to the dimension of col(p ) S and since S is k- dimensional, we see that the rank of P is k Problem If V is the subspace spanned by (,,, ) and (,,, ), find (a) a basis for the orthogonal complement V Answer: Consider the matrix [ A By construction, the row space of A is equal to V Therefore, since the nullspace of any matrix is the orthogonal complement of the row space, it must be the case that V nul(a) The matrix A is already in reduced echelon form, so we can see that the homogeneous equation A x is equivalent to x x x 4 x Therefore, the solutions of the homogeneous equation are of the form x + x 4, so the following is a basis for nul(a) V :, (b) the projection matrix P onto V Answer: From part (a), we have that V is the row space of A or, equivalently, V is the column space of B A T

2 Therefore, the projection matrix P onto V col(b) is P B(B T B) B T A T (AA T ) A Now, so Therefore, B T B AA T [ (AA T ) [ [, P A T (AA T ) A [ [ [ (c) the vector in V closest to the vector b (,,, ) in V Answer: The closest vector to b in V will necessarily be the projection of b onto V Since b is perpendicular to V, we know this will be the zero vector We can also doublecheck this since the projection of b onto V is P b Problem Find the best line C + Dt to fit b 4,,,, at times t,,,, Answer: If the above data points actually lay on a straight line C + Dt, we would have 4 [ C D

3 Call the matrix A and the[ vector on the right-hand side b Of course this system is inconsistent, C but we want to find x such that A x is as close as possible to D b As we ve seen, the correct choice of x is given by x (A T A) A T b To compute this, first note that A T A [ [ 5 Therefore, and so (A T A) [ 5 x (A T A) A T b [ 5 [ 5 [ 4 5 [ [ 5 8 Therefore, the best-fit line for the data is 4 5 t Here are the data points and the best-fit line on the same graph:

4 4 Problem 4 Find the best straight-line fit to the following measurements, and sketch your solution: y at t, y at t, y at t, y 5 at t Answer: As in Problem, if the data actually lay on a straight line y C + Dt, we would have [ C D 5 Again, this system is not solvable, but, if A is the matrix and b is the vector on the right-hand side, then we want to find x such that A x is as close as possible to b This will happen when x (A T A) A T b Now, A T A [ [ 4 6 To find (A T A), we want to perform row operations on the augmented matrix [ 4 6 so that the identity matrix appears on the left To that end, scale the first row by 4 and subtract times the result from row : [ / /4 5 / Now, scale row by 5 and subtract half the result from row : [ / / / /5 Therefore, (A T A) [ / / / /5 4

5 and so x (A T A) A T b [ / / / /5 [ / / / /5 [ / /5 Therefore, the best-fit line for the data is [ [ 6 5 y 5 t Here s a plot of both the data and the best-fit line: Problem 5 Suppose that instead of a straight line, we fit the data in Problem 4 (ie # above) by a parabola y C + Dt + Et In the inconsistent system A x b that comes from the four measurements, what are the coefficient matrix A, the unknown vector x, and the data vector b? For extra credit, actually determine the best-fit parabola Answer: Since the data hasn t changed, the data vector b will be the same as in the previous problem If the data were to lie on a parabola C + Dt + Et, then we would have that C D, E 4 5 so A is the matrix above and x is the vector next to A on the left-hand side To actually determine the best-fit parabola, we just need to find x such that A x is as close as possible to b This will be the vector x (A T A) A T b 5

6 Now, A T A To find (A T A), we want to use row operations to convert the left-hand side of this augmented matrix to I: First, scale row by 4 row : and subtract twice the result from row and six times the result from / / /4 5 5 / 5 9 / Next, subtract row from row, scale row by 5 and subtract half the result from row : / / / /5 4 Finally, scale row by 4 and subtract the result from rows and : / / /4 / 9/ /4 /4 /4 /4 Therefore, and so (A T A) / / /4 / 9/ /4 /4 /4 /4 x (A T A) A T b / / /4 / 9/ /4 /4 /4 /4 / / /4 / 9/ /4 /4 /4 /4 / /

7 Thus, the best-fit parabola is y 5 t + t 5 t, which is the same as the best-fit line! 6 Problem 44 If Q and Q are orthogonal matrices, so that Q T Q I, show that Q Q is also orthogonal If Q is rotation through θ and Q is rotation through φ, what is Q Q? Can you find the trigonometric identities for sin(θ + φ) and cos(θ + φ) in the matrix multiplication Q Q? Answer: Note that (Q Q ) T (Q Q ) Q T Q T Q Q Q T IQ Q T Q I, since both Q and Q are orthogonal matrices Therefore, the columns of Q Q are orthonormal Moreover, since both Q and Q are square and must be the same size for Q Q to make sense, it must be the case that Q Q is square Therefore, since Q Q is square and has orthonormal columns, it is an orthogonal matrix If Q is rotation through and angle θ, then, as we ve seen, [ cos θ sin θ Q sin θ cos θ Likewise, if Q is rotation through and angle φ, then [ cos φ sin φ Q sin φ cos φ With these choices of Q and Q, if x is any vector in the plane R, we see that Q Q x Q (Q x), meaning that x is first rotated by an angle φ, then the result is rotated by an angle θ Of course, this is the same as rotating x by an angle θ + φ, so Q Q is precisely the matrix of the transformation which rotates the plane through an angle of θ + φ On the one hand, we know that [ [ [ cos θ sin θ cos φ sin φ cos θ cos φ sin θ sin φ cos θ sin φ + sin θ cos φ Q Q sin θ cos θ sin φ cos φ sin θ cos φ cos θ sin φ sin θ sin φ + cos θ cos φ On the other hand, the matrix which rotates the plane through an angle of θ + φ is precisely [ cos(θ + φ) sin(θ + φ) sin(θ + φ) cos(θ + φ) Hence, it must be the case that [ [ cos(θ + φ) sin(θ + φ) cos θ cos φ sin θ sin φ cos θ sin φ + sin θ cos φ sin(θ + φ) cos(θ + φ) sin θ cos φ cos θ sin φ sin θ sin φ + cos θ cos φ This implies the following trigonometric identities: cos(θ + φ) cos θ cos φ sin θ sin φ sin(θ + φ) cos θ sin φ + sin θ cos φ 7

8 7 Problem 46 Find a third column so that the matrix / / 4 Q / / 4 / / 4 is orthogonal It must be a unit vector that is orthogonal to the other columns; how much freedom does this leave? Verify that the rows automatically become orthonormal at the same time Answer: Let q / / / and q q, q and q, q then we have that / 4 / 4 / 4 q q, q a + b + c If q q, q a/ + b/ + c/ q, q a/ 4 + b/ 4 c/ 4 a b c such that q, Multiplying the second line by and the third line by 4, we get the equivalent system a + b + c a + b + c a + b c From the second line we have that b a c and so, from the third line, a b + c ( a c) + c a + 5c Thus a 5c, meaning that b a c ( 5c) c 4c Therefore a + b + c ( 5c) + (4c) + c 4c, meaning that c ±/ 4 Thus, we see that q a b c 5c 4c c ± 5/ 4 4/ 4 / 4 Therefore, there are two possible choices; one of them gives the following orthogonal matrix: Q / / 4 5/ 4 / / 4 4/ 4 / / 4 / 4 It is straightforward to check that each row has length and is perpendicular to the other rows 8

9 [ [ 4 8 Problem 4 What multiple of a should be subtracted from a to make [ 4 the result orthogonal to a? Factor into QR with orthonormal vectors in Q Answer: Let s do Gram-Schmidt on { a, a } First, we let v a a [ [ / / Next, w a v, a v [ 4 4/ [ / / [ 4 [ [ By construction, w is orthogonal to a, so we see that we needed to subtract times a from a to get a vector perpendicular to a Now, continuing with Gram-Schmidt, we get that v w w [ Therefore, if A [ a a and Q [ v v, then A QR [ / / where R [ a, v a, v a, v [ 9 Problem 48 If A QR, find a simple formula for the projection matrix P onto the column space of A Answer: If A QR, then A T A (QR) T (QR) R T Q T QR R T IR R T R, since Q is an orthogonal matrix (meaning Q T Q I) Thus, the projection matrix P onto the column space of A is given by P A(A T A) A T QR(R T R) (QR) T QRR (R T ) R T Q T QQ T (provided, of course, that R is invertible) Problem 4 (a) Find a basis for the subspace S in R 4 spanned by all solutions of x + x + x x 4 9

10 Answer: The solutions of the given equation are, equivalently, solutions of the matrix equation x [ x x, x 4 so S is the nullspace of the 4 matrix A [ Since A is already in reduced echelon form, we can read off that the solutions to the above matrix equation are the vectors of the form x + x + x 4 Therefore, a basis for nul(a) S is given by,, (b) Find a basis for the orthogonal complement S Answer: Since S nul(a), it must be the case that S is the row space of A Hence, the one row of A gives a basis for S, meaning that the following is a basis for S : (c) Find b in S and b in S so that b + b b (,,, ) Answer: For any b S, we know that b is a linear combination of elements of the basis for S that we found in part (a) In other words, b a + b + c for some choice of a, b, c R Also, if b S, then b is a multiple of the basis vector for S we found in part (b) Thus, b d

11 for some d R Therefore, b a + b + c + d or, equivalently, a b c d To solve this matrix equation, we just do elimination on the augmented matrix Add row to row : Next, add row to row : Finally, subtract row from row 4: Therefore, 4d, so d Hence, 4 c + d c +, so c In turn, b + c + d b + + b + 5, meaning b Finally, a b + c + d a + + a +,

12 so a Therefore, and b + b + / / / / / / / / (Bonus Problem) Problem 44 Find the fourth Legendre polynomial It is a cubic x + ax + bx + c that is orthogonal to, x, and x over the interval x Answer: We can find the fourth Legendre polynomial in the same style as Strang finds the third Legendre polynomial on p 85: v 4 x, x, x, x x x, x x, x ( x, x x ) () Now, we just compute each of the inner products in turn: Therefore, () becomes, x, x, x x, x x, x x, x x dx dx x 4 dx 5 x dx (x 5 x ( x v 4 x /5 / x Therefore, the fourth Legendre polynomial is x 5 x ) dx ) dx 8 45 ( x ) x 5 x

### 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

### 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

### 18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points.

806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 2-06 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are right-hand-sides b for which A x = b has no solution (a) What

### 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

### 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

### 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

### 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

### LINEAR ALGEBRA. September 23, 2010

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

### 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.

### 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

### ( ) 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

### 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

### 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,

### 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,

### 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

### 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

### 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

### 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?

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### 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

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### 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

### 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,

### 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

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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).

### 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

### 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

### 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

### 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

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC

MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC DR. LESZEK GAWARECKI 1. The Cartesian Coordinate System In the Cartesian system points are defined by giving their coordinates. Plot the following points:

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### 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

### Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

### System of First Order Differential Equations

CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions

### 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

### 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

### Systems of Linear Equations

Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear

### 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

### 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

### 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

### 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

### 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 +

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### 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 =

### University of Tampere Computer Graphics 2013 School of Information Sciences Exercise 2 7.2.2013

University of Tampere Computer Graphics 2013 School of Information Sciences Exercise 2 7.2.2013 1. We can get the minmum of parabola f(x) = ax 2 + bx + c,a > 0,, when f (x) = 0. In this case x has the

### 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

### 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

### Linear Algebra Methods for Data Mining

Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 Lecture 3: QR, least squares, linear regression Linear Algebra Methods for Data Mining, Spring 2007, University

### 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

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### 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)

### 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

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

### 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

### 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

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007

KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and

### 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

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### 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

### 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

### 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

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

### The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

### Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### 1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### ( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### Applied Linear Algebra

Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors Chia-Hui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### 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

### ME 115(b): Solution to Homework #1

ME 115(b): Solution to Homework #1 Solution to Problem #1: To construct the hybrid Jacobian for a manipulator, you could either construct the body Jacobian, JST b, and then use the body-to-hybrid velocity

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### 8. Linear least-squares

8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.