Examination in TMA4110/TMA4115 Calculus 3, August 2013 Solution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Examination in TMA4110/TMA4115 Calculus 3, August 2013 Solution"

Transcription

1 Norwegian University of Science and Technology Department of Mathematical Sciences Page of Examination in TMA40/TMA45 Calculus 3, August 03 Solution 0 0 Problem Given the matrix A a) Write the solution set of the matrix equation Ax 8 in parametric vector form. Solution. We begin by reducing the augmented matrix of the equation to its reduced echelon form We see that the last column does not contain a pivot position, so the equation is consistent. We furthermore see that the first and the second columns contain pivot position, and that the third and forth do not, so we choose x 3 and x 4 to be free variables. The reduced echelon form of the augmented matrix of the equation corresponds to the system x + x 3 x + 5x 3 + x 4 3 It follows that the complete solution of the equation is x s 0 x x x 3 3 5s t s s 5 + t 0 x 4 t where s and t are free parameters.

2 Page of b) Find an orthonormal basis for the row space Row(A). Solution. We begin by reducing A to its reduced echelon form We see that the first and second rows contain pivot position, and that the third does not. It follows that the first row v [ 0 0 ] of the reduced echelon form of A and the second row v [ 0 5 ] of the reduced echelon form of A form a basis of Row(A). We next use the Gram-Schmidt process in order to get an orthogonal basis of Row(A). Let u v [ 0 0 ] and u v v u u [ 0 5 ] 0 [ ] [ ] 0 0. u u 5 Then {u, u } is an orthogonal basis of Row(A). Finally, we normalize u and u in order to get an orthonormal basis of Row(A). Let x u [ ] u and x u [ ] u 0. Then {x, x } is an orthonormal basis for the row space Row(A). Problem [ ] 3 a) Diagonalize the matrix A 3 matrix D such that A P DP ). (that is, find an invertible matrix P and a diagonal Solution. The characteristic polynomial of A is det(a λi) λ 3 3 λ ( λ) 3 λ λ 8, and the eigenvalues of A are λ ± [ ] 3 3 A 4I. It follows that v 3 3 eigenvalue 4. [ ] ± 6 { 4. is an eigenvector of A corresponding to the

3 [ ] 3 3 A + I. 3 3 It follows that v eigenvalue. Thus, if we let P b) Make a change of variable, [ ] [ ] [ ] 4 0 and D, then A P DP 0. [ x x ] B [ y y Page 3 of is an eigenvector of A corresponding to the ], that transforms the quadratic form x + 6x x + x into [ ] a quadratic [ ] form with no cross-product term (that is, find a matrix B x y such that if B, then x + 6x x + x ay + by for a suitable choice of x constants a and b). y Solution. The idea is to let B be an orthogonal matrix whose columns are eigenvectors of A. The eigenvectors v and v we found in a) are orthogonal to each other so we just have to normalize them. Since v v [ ], it follows that if we let B P, then B is an orthonormal matrix, so B T B (in fact, B T B B). Thus, if we let [ ] [ ] [ ] x y (y + y ) B, then (y y ) x y x + 6x x + x (We could also just had let ) [ ] (y + y ) + 6 (y + y ) [ ] (y y ) + (y y ) y + y y + y + 3y 3y + y y y + y 4y y. [ x x ] P [ y y ] [ ] y + y, then y y x + 6x x + x (y + y ) + 6(y + y )(y y ) + (y y ) y + y y + y + 6y 6y + y y y + y 8y 4y. Problem 3 Let y (t) t and y (t) t ln(t) be two solutions of the differential equation t y ty + y 0 on the interval (0, ).

4 Page 4 of a) Show that the set {y (t), y (t)} is a fundamental set of solutions for the above equation on the interval (0, ). Solution. Let us first check that y and y are solutions to the differential equation t y ty + y 0 (this is not really necessary, since it is stated in the problem that they are). We have that y (t) and that y (t) 0, so t y (t) ty (t) + y (t) 0 t + t 0, and y (t) ln(t) +, y (t) /t, so t y (t) ty (t) + y (t) t t ln(t) t + t ln(t) 0. Thus, y and y are solutions to the differential equation t y ty + y 0. To show that {y (t), y (t)} is a fundamental set of solutions for the equation, we compute the Wronskian of y and y. W (y, y )(t) y (t)y (t) y (t)y (t) t(ln(t) + ) t ln(t) t 0 for all t (0, ), so {y (t), y (t)} is a fundamental set of solutions for the differential equation t y ty + y 0. Alternatively, we could show directly that y and y are linearly independent on the interval (0, ), for suppose that c and c are constants such that c y (t)+c y (t) 0 for all t (0, ), then we get by evaluating at t that c 0 (since y () and y () 0), and then by evaluating at t exp() that c 0 (since y (exp()) exp() 0). This shows that y and y are linearly independent on the interval (0, ), and it follows that {y (t), y (t)} is a fundamental set of solutions for the differential equation t y ty + y 0. b) Find the general solution to the differential equation t y ty +y t, where 0 < t <. Solution. We will first find a particular solution by using variations of parameters. So we define a function y on the interval (0, ) by letting y(t) v (t)y (t) + v (t)y (t) for each t (0, ) where v and v are twice differentiable functions to be determined below. Then y (t) v (t)y (t) + v (t)y (y) + v (t)y (t) + v (t)y (t). We will assume that v (t)y (t) + v (t)y (y) 0 for all t (0, ). Then y (t) v (t)y (t) + v (t)y (t), y (t) v (t)y (t) + v (t)y (t) + v (t)y (t) + v (t)y (t), and t y (t) ty (t) + y(t) t (v (t)y (t) + v (t)y (t) + v (t)y (t) + v (t)y (t)) t(v (t)y (t) + v (t)y (t)) + v (t)y (t) + v (t)y (t) v (t)(t y (t) ty (t) + y (t)) + v (t)(t y (t) ty (t) + y (t)) + v (t)t y (t) + v (t)t y (t) v (t)t + v (t)t (ln(t) + ).

5 Page 5 of Thus, if v (t)y (t) + v (t)y (y) v (t)t + v (t)t ln(t) 0 and v (t)t + v (t)t (ln(t) + ) t for all t (0, ), then y is a solution to the differential equation t y ty + y t on the interval (0, ). It is easy to see that for any t (0, ), the system v (t)t + v (t)t (ln(t) + ) t v (t)t + v (t)t ln(t) 0 has a unique solution and that this solution is v (t) /t and v (t) ln(t)/t. It follows that if we let v (t) v (t)dt ln(t)/t dt (ln(t)), and v (t) v (t)dt /t dt ln(t), then y(t) y (t)v (t) + y (t)v (t) t(ln(t)) + t(ln(t)) t(ln(t)) is a solution to the differential equation t y ty + y t on the interval (0, ). It follows that the general solution to the differential equation t y ty + y t on the interval (0, ) is where c and c are constants. y(t) t(ln(t)) + c t + c t ln(t) Problem 4 0 a) The matrix A 0 has 3 eigenvalues. One eigenvalue is + i, and 0 + i i is an eigenvector corresponding to + i. Find the other two eigenvalues and a corresponding eigenvector for each of these eigenvalues.

6 Page 6 of + i Solution. Since all the entries of A are real, and i is an eigenvector of A corresponding i to + i, it follows by complex conjugation that + i is an eigenvector of A corresponding to i. The characteristic polynomial of A is λ 0 det(a λi) λ 0 ( λ)( λ)( λ) + 0 λ λ 3 4λ 5λ. It follows that 0 is an eigenvalue of A. In order to find a corresponding eigenvector, we reduce A to an echelon form We see that is an eigenvector of A corresponding to the eigenvalue 0. b) Three tanks contain salt water. Tank holds 0 litres, tank holds 5 litres, and tank 3 holds 0 litres. Salt water flows from tank to tank at a rate of 0 litres per second, and from tank to tank 3 at a rate of 0 litres per second, and from tank 3 to tank at a rate of 0 litres per second. Suppose that initially tank contains 0 grammes of salt, tank contains 6 grammes of salt, and tank 3 contains 4 grammes of salt. Suppose also that the tanks are stirred so that the salt in each tank is evenly distributed. How much salt does each of the three tanks contains after π/ seconds? Solution. Let x (t) be the amount of salt in tank at time t, let x (t) be the amount of salt in tank at time t, and let x 3 (t) be the amount of salt in tank 3 at time t. Then x (t) x (t) + x 3 (t) x (t) x (t) x (t) x 3(t) x (t) x 3 (t)

7 Page 7 of and x (0) 0, x (0) 6, x 3 (0) 4. Thus, it follows from a) that x (t) + i i x (t) c i e ( +i)t + c + i e ( i)t + c 3 x 3 (t) where c, c and c 3 are constants satisfying + i i 0 c i + c + i + c To solve the last equation, we reduce the augmented matrix of the equation to its reduced echelon form. + i i 0 4 i + i 6 i + i 6 0 i i 8 i 4 + i i i 3 + i + i 0 i i 8 i 0 i i i Thus c c and c 3 4, and x (t) + i i 8 x (t) i e ( +i)t + + i e ( i)t +4 e t cos(t)+ e t sin(t)+ 4 x 3 (t) It follows that x (π/) 8 e π, x (π/) 4+e π and that x 3 (π/) 8, so tank contains 8 e π grammes of salt after π/ seconds, tank contains 4 + e π grammes of salt after π/ seconds, and tank 3 contains 8 grammes of salt after π/ seconds. Problem 5 You do not have to give reasons for your answers for this problem. a) For each of the following 4 statements, determine whether it is true or not.. The equation x x ( + i) is linear. True.

8 Page 8 of. The differential equation y + 6y e 4t + 3 sin(4t) is linear. True. 3. The differential equation t y (t) + 3ty (t) 3y(t) 0 is linear. True. 4. The transformation T from R to R given by T (x) x + x is linear. False. We have for example that T () 6 4 T (). b) For each of the following 4 statements, determine whether it is true or not.. The three lines x + x 3, x + x 3 and x + x 3 have exactly one point in common. 0 0 True. The matrix 0 3 can be reduced to so (0, 3, ) is 0 the only point which belong to all three lines If a, a and b, then b belongs to Span{a, a }. 0 4 False. The matrix can be reduced to 0 4 which has a pivot position in the last column, so b does not belong to Span{a, a } If v 0, v 0 and v 3 0 0, then Span{v, v, v 3 } R 4. 0 False. A set of three vectors cannot generate R The following vectors are linearly independent: 4, 6 and False c) For each of the following 4 statements, determine whether it is true or not.. If A is an m n matrix, B is an n m matrix, and AB 0, then we must either have that A 0 or that B 0. False. We have for example that [ 0 ] [ ] 0 0.

9 Page 9 of. If C and D are n n-matrices, then it must be the case that (C + D)(C D) C D. False. We have for example that ([ ] 0 + [ ]) ([ ] 0 0 and [ ] 0 [ ] 0 [ ]) 0 [ ] 0 3. If E is an invertible matrix, then (E) E. False. (E) E. 4. If F is an matrix and the equation F x must be invertible. [ ] [ 0 0 [ ] ] [ ] 0 0 [ ] 0 [ ] 0 0 [ ] has a unique solution, then F True. If the equation F x has a unique solution, then Nul(F ) {0} (because [ ] [ ] if u Nul(F ), u 0 and F x, then F (x + u) ), and it then follows from the invertible matrix theorem that F is invertible. d) Let A be an n n matrix and k a scalar. For each of the following 4 statements, determine whether it is true or not.. det(ka) k n det(a). True. ka can be produced by multiplying each of the n rows of A by k, so it follows from Theorem 3 in Section 3. that det(ka) k n det(a).. If det(a), then det(a ) 4. True. det(a ) det(a) det(a) by Theorem 6 in Section det(a T ) det(a). True. See Theorem 5 in Section If A is invertible, then det(a) det(a ) 0. False. det(a) det(a ) det(a) det(a) 0. e) Let V be a vector space different from the zero vector space, and let v, v,..., v p be vectors in V. For each of the following 4 statements, determine whether it is true or not.. If {v, v,..., v p } is linearly independent, then it must be the case that {v, v,..., v p } is a basis for V. False. It is only true if dim(v ) p.

10 Page 0 of. If Span{v, v,..., v p } V, then some subset of {v, v,..., v p } is a basis for V. True. See Theorem 5 in Section If dim(v ) p, then {v, v,..., v p } cannot be linearly independent. False. If {v, v,..., v p } is a basis for V, then {v, v,..., v p } is linearly independent. 4. If dim(v ) p, then {v, v,..., v p } must be a basis for V. False. {v, v,..., v p } is only a basis for V if it is linearly independent or Span{v, v,..., v p } V. f) Let A be an n n matrix. For each of the following 4 statements, determine whether it is true or not.. If A is invertible and is an eigenvalue of A, then must also be an eigenvalue of A. True. If Av v, then v (A A)v A v.. If v is an eigenvector of A, then v must also be an eigenvector of A. True. If Av λv, then A v λav λ v. 3. If A has fewer than n distinct eigenvalues, [ ] then A[ cannot ] be diagonalizable. 0 0 False. is the only eigenvalue of, and is a diagonale matrix and 0 0 therefore diagonalizable. 4. If every vector in R n can be written as a linear combination of eigenvectors of A, then A must be diagonalizable. True. If every vector in R n can be written as a linear combination of eigenvectors of A, then R n has a basis consisting of eigenvectors of A, and it then follows from Theorem 5 in Section 5.3 that A is diagonalizable. g) Let v and u be vectors in R n and let W be a subspace of R n. For each of the following 4 statements, determine whether it is true or not.. The distance between v and u is u v. True.. If {u, v} is linearly independent, then [ ] u and [ v ] must be orthogonal to each other. False. We have for example that and are linearly independent but not 0 orthogonal to each other. 3. If v coincides with its orthogonal projection onto W, then v must belong to W. True. The orthogonal projection of v onto W belongs to W so if it coincides with v, then v must belong to W.

11 Page of 4. No vector in R n can belong to both W and W (the orthogonal complement of W ). False. 0 belongs to both W and W. h) Let A be an n n matrix. For each of the following 4 statements, determine whether it is true or not.. If A T A and if u and v are vectors in R n which satisfy that Au 5u and Av v, then u and v must be orthogonal to each other. True. See Theorem in Section 7... If A P DP T where P T P and D is a diagonal matrix, then A must be symmetric. True. A T (P DP T ) T P DP T A because D T D. 3. If A is symmetric and all the eigenvalues of A are positive, then the quadratic form x x T Ax must be positive definite. True. See Theorem 5 in Section If A is symmetric and λ is an eigenvalue of A, then the dimension of the eigenspace of A corresponding to λ must be equal to the multiplicity of λ as a root of the characteristic polynomial of A. True. See Theorem 3 in Section 7..

1 Eigenvalues and Eigenvectors

1 Eigenvalues and Eigenvectors Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

More information

Orthogonal Diagonalization of Symmetric Matrices

Orthogonal Diagonalization of Symmetric Matrices MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

More information

Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

More information

Similarity and Diagonalization. Similar Matrices

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

More information

1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

More information

Examination paper for TMA4115 Matematikk 3

Examination paper for TMA4115 Matematikk 3 Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99

More information

Chapter 6. Orthogonality

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

More information

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

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

More information

Inner Product Spaces and Orthogonality

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,

More information

Notes on Jordan Canonical Form

Notes on Jordan Canonical Form Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main

More information

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

MATH 551 - APPLIED MATRIX THEORY

MATH 551 - APPLIED MATRIX THEORY 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

More information

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

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

More information

MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

More information

C 1 x(t) = e ta C = e C n. 2! A2 + t3

C 1 x(t) = e ta C = e C n. 2! A2 + t3 Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

More information

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

More information

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

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?

More information

[1] Diagonal factorization

[1] Diagonal factorization 8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

More information

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1 (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

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

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

More information

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A = Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

More information

Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

More information

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

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,

More information

Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Presentation 3: Eigenvalues and Eigenvectors of a Matrix Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

Similar matrices and Jordan form

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

More information

4.1 VECTOR SPACES AND SUBSPACES

4.1 VECTOR SPACES AND SUBSPACES 4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following

More information

(January 14, 2009) End k (V ) End k (V/W )

(January 14, 2009) End k (V ) End k (V/W ) (January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t) Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

Inner products on R n, and more

Inner products on R n, and more Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

More information

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d. DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of Matrix-Valued Functions of a Real Variable

More information

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 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

More information

[: : :] [: :1. - B2)CT; (c) AC - CA, where A= B= andc= o]l [o 1 o]l

[: : :] [: :1. - B2)CT; (c) AC - CA, where A= B= andc= o]l [o 1 o]l Math 225 Problems for Review 1 0. Study your notes and the textbook (Sects. 1.1-1.5, 1.7, 2.1-2.3, 2.6). 1. Bring the augmented matrix [A 1 b] to a reduced row echelon form and solve two systems of linear

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information

1 VECTOR SPACES AND SUBSPACES

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

More information

NOTES on LINEAR ALGEBRA 1

NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

More information

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

More information

Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

More information

Sec 4.1 Vector Spaces and Subspaces

Sec 4.1 Vector Spaces and Subspaces Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

More information

1 Introduction to Matrices

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

More information

Linear Algebra Review. Vectors

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

More information

22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

More information

MATH10212 Linear Algebra B Homework 7

MATH10212 Linear Algebra B Homework 7 MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

More information

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

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

More information

Matrix Methods for Linear Systems of Differential Equations

Matrix Methods for Linear Systems of Differential Equations Matrix Methods for Linear Systems of Differential Equations We now present an application of matrix methods to linear systems of differential equations. We shall follow the development given in Chapter

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,

More information

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above: Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

More information

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

More information

Chapter 20. Vector Spaces and Bases

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

More information

Higher Order Equations

Higher Order Equations Higher Order Equations We briefly consider how what we have done with order two equations generalizes to higher order linear equations. Fortunately, the generalization is very straightforward: 1. Theory.

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

Eigenvalues and Eigenvectors

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

More information

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.

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

More information

Inner Product Spaces. 7.1 Inner Products

Inner Product Spaces. 7.1 Inner Products 7 Inner Product Spaces 71 Inner Products Recall that if z is a complex number, then z denotes the conjugate of z, Re(z) denotes the real part of z, and Im(z) denotes the imaginary part of z By definition,

More information

System of First Order Differential Equations

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

More information

Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

More information

Inner Product Spaces

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

More information

4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3

4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3 MT Notebook Fall / prepared by Professor Jenny Baglivo c Copyright 9 by Jenny A. Baglivo. All Rights Reserved. Contents MT Notebook. Eigenvalues and Eigenvectors................................... Definitions;

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

4.6 Null Space, Column Space, Row Space

4.6 Null Space, Column Space, Row Space NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

More information

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

Applied Linear Algebra I Review page 1

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

More information

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

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

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

1 Sets and Set Notation.

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

More information

Linear Algebra Problems

Linear Algebra Problems Math 504 505 Linear Algebra Problems Jerry L. Kazdan Note: New problems are often added to this collection so the problem numbers change. If you want to refer others to these problems by number, it is

More information

Definition: A square matrix A is block diagonal if A has the form A 1 O O O A 2 O A =

Definition: A square matrix A is block diagonal if A has the form A 1 O O O A 2 O A = The question we want to answer now is the following: If A is not similar to a diagonal matrix, then what is the simplest matrix that A is similar to? Before we can provide the answer, we will have to introduce

More information

Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 Spanning and Independence Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

More information

Math 312 Homework 1 Solutions

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

More information

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 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)(

More information

MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

More information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

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

More information

The Characteristic Polynomial

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

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Brief Introduction to Vectors and Matrices

Brief Introduction to Vectors and Matrices CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

More information

2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

1.5 SOLUTION SETS OF LINEAR SYSTEMS

1.5 SOLUTION SETS OF LINEAR SYSTEMS 1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as

More information

LINEAR ALGEBRA AND MATRICES M3P9

LINEAR ALGEBRA AND MATRICES M3P9 LINEAR ALGEBRA AND MATRICES M3P9 December 11, 2000 PART ONE. LINEAR TRANSFORMATIONS Let k be a field. Usually, k will be the field of complex numbers C, but it can also be any other field, e.g., the field

More information

Solutions to Review Problems

Solutions to Review Problems Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a

More information

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

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

More information

BASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM

BASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM BASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM JORDAN BELL Abstract. This paper gives a basic introduction to the Jordan canonical form and its applications. It looks at the Jordan canonical

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

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

More information