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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 Math 225 Problems for Review 1 0. Study your notes and the textbook (Sects , 1.7, , 2.6). 1. Bring the augmented matrix [A 1 b] to a reduced row echelon form and solve two systems of linear equations (SLEs) Ax = b, Ax = 0, indicating basic and free unknowns, where [: : :] [: :1 (a) A = and b = [1,1,3IT; (b) A = and b = [l,l,ojt. 2. Write down solutions of SLE's of Problem 1 in parametric vector form. 3. Explain the relation between solution sets of SLE's Ax = b and Ax = 0 and give an example of your choice. 4. (a) Let xl = [l, 0, 1IT and x:! = [I, 2, 3JT be solutions of a SLE Ax = b. Find (if possible) one more solution. (b) Let xl = [I, 1,1, 1IT and xz = [3,1,2, 5IT be solutions of a SLE Ax = b. Find (if possible) one more solution. 5. Compute (if possible) the following matrices (a) A(B f C) + 2C(BT - A); (b) (AZoo7 - B2)CT; (c) AC - CA, where A= B= andc= o]l [o 1 o]l [o 0 6. (a) Give the definition of linearly independent (and dependent) vectors vl,...,v, E Rn. (b) Explain how to find out whether given vectors al,...,a, E Rm are linearly independent and give an example of your choice. 7. Determine whether following vectors are linearly independent (a)al=[1,2,3,o,l]t,a2=[1,2,3,1,1]t,a~=[3,2,3,1,1]t,aq=[2,4,6,1,3]t; (b) a, = [I, 2,3,0, ljt, a2 = [1,2,3,1, ljt, a3 = [3,2,3,1, ljt; (c) Columns of the matrices of Problem la and lb. 8. Let B = [bl... b,], where bl,..., b, are columns of matrix B. Show that AB = [Abl... Ab,] Suppose that A is an m x n matrix and B is an n x p matrix. (a) Show that if columns of B are linearly dependent then columns of the product AB are also linearly dependent. (Hint: Use Problem 8.) (b) Show that if columns of the product AB are lmearly independent then columns of B are also linearly independent. 10. (a) Give the definition of the span Spau(vl,...,v,) of vectors vl,...,v, E Rn. (b) Determine whether vector a4 = [2,4,6,1, 3JT lies in the span of a1 = [I, 2,3,0, ljt, a2 = [I, 2,3,1, 1IT, a3 = [3,2,3,1, 1IT. 11. Let vl,...,v, E Rm. Explain how to determine if Span(vl,..., v,) = Rm and give an example of your choice. 12. Give the definition of the identity matrix I,, of size n x n and prove equalities I,A = A, BI, = B, where A has size n x m and B has. size m x n. 13. (a) Give the definition of an inverse of a matrix. (b) Show that an inverse of a matrix is unique. 14. Let A, B be invertible matrices. (a) Show that A-', AT are invertible and (A-')-I = A, (AT)-' = (A-l)T. (b) Show that AB is invertible and (AB)-' = B-'A-'.

2 A 15. Suppose that E is an elementary matrix of one of three types E,, E,(c), E,j(c). Are ET, E-' elementary matrices? What do ET, E-' look like? 16. Determine if matrix A is row equivalent to B if [ (b) A= :I :I [ andb= :I. 17. Find the inverse A-' [(if any) and elementary ] matrices El,..., Ek such that Ek... EIA = In if (a) A 1 ; ) ( ) = [ 5: Formulate four equivalent conditions that are all equivalent to the existence of the inverse A-' of an n x n matrix A. Sketch the proof of equivalence of these conditions. 19. Explain the Leontief input-output model and its main production (matrix) equation x = Cx + d (or (In - C)X = d). 20. (a) Determine the production vector - x to satisfy the final demand d = [20, 30IT if the consumption matrix is C = (b) Determine the - production - vector x to satisfy the final demand d = [40, 60IT if the consumption I, matrix is C = I ::: ::; L~ ~ (c) Determine the production vector x to satisfy the final demand d = [loo, 100, 100IT if the consumption matrix is C =

3 Math 225 Problems for Review 2 0. Study your notes and the textbook (Sects , , ) 1. (a) Give the definition of the determinant det A of an n x n matrix A, where n > 1. (h) Show that if A = (aij),,, integer. is an n x n matrix with integer entries ad then det A is also an 2. (a) Explain what happens to det A when an elementary row (column) operation is applied to A (b) Evaluate det E if E is an elementary matrix. (c) Suppose that B is a matrix which is row equivalent to A and det A = 2. Find (if possible) det B. 3. Evaluate det A if A is the following matrix. (a) A = Evaluate det A if (a) A = n... n b a... a a a a... n n... nxn a a... a b 5. Let al, az,...,a, be columnsof an n x n matrix A = [a, a2... a,] and det A = 2. Evaluate the following determinants (a) det[a, al a2... a,_l]; (b) det[al 2az... (n - l)a,-l na,]; (c) det[al a1 + az a2 + as... a,_, + aj; nxn 6. Find (if possible) detza, det(-b), det(a + B), det(aba2b2), det(adja), det(adjb) if A, B are matrices of size n x n with det A = 1 and det B = Show that (a) An n x n matrix A is invertible if and only if det A # 0. (b) If A, B are n x n matrices, then det AB = det A det B. 8. (a) Write down formulas for the adjoint adj A, and the inverse A-I of a matrix A. (b) Explain and prove the false expansion formula and use this formula to show that adja.a=a.adja=deta.i,. 9. Find (if possible) the adjoint adj A if (a) A = Explain Cramer's rule and, using this rule (if possible), solve SLE Ax = b, where andb=[l,o,1it.

4 11. Give the definition of a vector space. Check this definition for Rn, Rmxn, P n, P, C[a, bl 12. (a) Give the definition of a subspace of a vector space V. (b) Show that if ul,..., vk are vectors in a vector space V then Span(vI,..., uk) is a subspace of v. 13. (a) Is S = {(xi,x~,x~,x~)~ I XI + 22 = x3 + xq} a subspace of R4? If so, find a basis and dim S. (b) Same problem for the subset S = {(XI, x2,x3, x4)t ( sl + zz + 3 = x3 + x4) of R4. T (c) Same problem for S = {(xi,x~,x~,x~) I zl + xz = x3 + x4,xl + x3 = xz + 24). 14. (a) Is S = {p(t) ( p(0) = 0) a subspace of P4? If SO, find a basis and dims. (b) Same problem for the subset S = {p(t) Lp(1) = p(2)} of PQ. (c) Same problem for the subset S = {p(t) I all coefficients of p(t) are integers} of PF, 15. Explain how to determine whether (a) n vectors UI,..., v, E IWm span Rm; (b) n vectors q,..., u, E Rm are linearly independent; (c) m vectors vl,..., v, E Rm are linearly independent Give examples of your choice. 16. (a) Give the definition of a basis and the dimension dim V of a vector space V. (h) Show that if V is a vector space and (bi,..., b,) is a basis for V then any m vectors, where m > n, are linearly dependent in V. 17. Suppose that B is a matrix in row echelon form. (a) Show that pivot columns of B form a basis for ColB. (b) Show that nonzero rows of B form a basis for RowB. 18. Suppose that V is a vector space and dimv = n. Prove that (a) If vl,...,u, are linearly independent vectors, then (q,... ;un) is a basis for V. (b) If Span(vl,...,vn) = V, then (ul,...,v,) is a basis for V. 19. Letul=(1,~,l,2)T,v2=(2,l,1,1)T,v~=(1r0,l,0)T,uq=(4,1,3,5)T (a) Extend (if possible) (ul, uz, us) to a basis of JR4. (b) Are vl, uz, vq linearly independent? (c) Is Span(vl,v~,u3,u4) = R4? 20. Let A be an rn x n matrix. Show that (a) dim ColA = dim RowA. (b) ranka t dim NulA = n. (c) SLE Ax = b has a solution if and only if b E ColA (d) ranka = rankat. 21. Find ranka, bases and dimensions of ColA, RowA, NulA if

5 Math 225 / Problems for Review 3 0. Study your notes and the textbook (Sects , , 6.5) 1. Give definitions of eigenvalues, eigenvectors and eigenspaces of an n x n matrix A. Show that (a) X is an eigenvalue of A if and only if det(a - XI,) = 0; (b) u is an eigenvector of A, corresponding to an eigenvalue A, if and only if u # 0 and u t Nul(A - XI,); (c) The eigenspace E(X) of A, corresponding to an eigenvalue A, is E(X) = Nul(A -XIn). [ i i Let A = (a) Which of are eigenvalues of A? J (b) Which of (1,1,1, I )~, (2,1,2, (O,O, 0, O)T are eigenvectors of A? 3. (a) Suppose A is a 3 x 3 matrix whose eigenvalues are -7,1,2. Find (if possible) eigenvalues of A (b) Let A2 = 0, where A is an n x n matrix. Show that if X is an eigenvalue of A, then X = Prove that an n x n matrix A is singular (that is, det A = 0) if and only if 0 is an eigenvalue of A.,I; 5. Diagonalize A (if possible), that is, find a representation of the form A = PDP-', where D is diagonal, for (a)-4=[o (,)A=[, ,I; = :I; I], 6. Let A be a matrix of Problem 5. Evaluate the product AIO1u, where v = (-2,2, 2)T, and compute products AIO'el, A'01e2, A10'e3, A'. 7. (a) State a condition that guarantees that an n x n matrix A is diagonalizable. (b) Give an example of an n x n matrix which is not diagonalizable. Prove your answer. 8. Let x, y be vectors in Rn. Show that (a) (Cauchy-Schwartz inequality) (x. yl 5 (/XI(. ((y((. (b) (Pythagorean law) If xly (x and y are orthogonal) then IJz + y((2 = ((x))~ +]lyj)2. 9. Let H be a subspace of Wn. State the definition of the orthogonal complement HL of H and show that HL is also a subspace, H n H I = {O) and dim H +dim HL = n. 10. (a) Explain the formulas Col(A)I = Nul(AT), Col(AT)I = Nul(A). (b) Find a basis and the dimension of the orthogonal complement H I of H if H is the subspace of R4 given by H = Span((1, -2,O, 3)T, (0,1,2; 11. (a) Show that if a, b E Rn, a # 0, then the (orthogonal) projection projab of b onto a is equal to %a. (b) Find the orthogonal projection of (1,2,2,~)~ onto (3,2,1, z )~. (c) Find the orthogonal projection of el - ez + ea onto el + es. 12. (a) Give definitions of orthogonal and orthonormal sets of vectors in Rn. (b) Prove that if (ul,..., uk) is an orthogonal set of nonzero vectors in Rn, then vectors 211,..., uk are linearly independent. 13. Show that if W = Span(ul,...,uk) is a subspace of Rn, y E Rn and (UI,...,uk) is an orthogonal basis of W, then the vector p = projw(y) = z u i his the following properties (a) p is the orthogonal projection of y onto W, that is, y = p + z, where z t WL; (b) the length lly - wll, where w t W, is minimal when w = p. 14. Compute projw(y) and minlly - wll over all w E W if (a) y = (1,1,2,3)T and W = Span(u~,uz,us), ul = (0,1,0, I)*, u2 = (1,0, -1,0)~, u3 = (I,& l,~)~; (b) y = (1,1,2)~ and W = Nul([l, 1, I]). 15. Give the definition of an orthogonal matrix Q of size n x n. Prove that (ul,...,u,) is an orthonormal basis for Rn if and only if U = [ul... un] is an orthogonal matrix. 16. State and prove the theorem on least squares solutions of a SLE Ax = b. 17. Find a least squares solution to the the folowing systems of linear equations: Xl + xz = 4 x1+z2 =2 (a) XI = 1 ; (b) XI - 22 = " = 1 XI + 2x7 - = Find a best least squares fit by a linear fnnction to the following data: y (a) 1 ; (1 m.

### 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

### University of Ottawa

University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)

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

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

### 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.

### 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

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### 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

### 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

### 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

### 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

### 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

### 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

### LINEAR ALGEBRA. September 23, 2010

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

### 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

### x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra-Lab 2

Linear Algebra-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4) 2x + 3y + 6z = 10

### 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

### 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

### 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

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

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

### 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

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Math 215 Exam #1 Practice Problem Solutions

Math 5 Exam # Practice Problem Solutions For each of the following statements, say whether it is true or false If the statement is true, prove it If false, give a counterexample (a) If A is a matrix such

### 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

### Math Practice Problems for Test 1

Math 290 - Practice Problems for Test 1 UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT. 3 4 5 1. Let c 1 and c 2 be the columns of A 5 2 and b 1. Show that b Span{c 1, c 2 } by 6 6 6 writing b as a linear

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### 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

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### 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

### Review: Vector space

Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space.

### 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

### 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

### 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

### 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

### 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

### APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

### 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)

### Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of

### Problems for Advanced Linear Algebra Fall 2012

Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are

### 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

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### 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

### 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

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### Harvard College. Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review

Harvard College Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu May 5, 2010 1 Contents Table of Contents 4 1 Linear Equations

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### 1 Determinants. Definition 1

Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

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

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

### 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

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### Lecture 10: Invertible matrices. Finding the inverse of a matrix

Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### 1 Orthogonal projections and the approximation

Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

### 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

### 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)(

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### 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

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### 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

### 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

### Matrices: 2.3 The Inverse of Matrices

September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.

### ( ) 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

### 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

### 5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES

5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES Definition 5.3. Orthogonal transformations and orthogonal matrices A linear transformation T from R n to R n is called orthogonal if it preserves

### Math 2331 Linear Algebra

2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

### 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

### 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

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### 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

### Algebra and Linear Algebra

Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### 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,

### MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?

MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All

### 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

### 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).

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### 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

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

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

### Determinants. Chapter Properties of the Determinant

Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

### 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

### CHARACTERISTIC ROOTS AND VECTORS

CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### 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

Advanced linear algebra M. Anthony, M. Harvey MT8, 798 Undergraduate study in Economics, Management, Finance and the Social Sciences This is an extract from a subject guide for an undergraduate course

### 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

### Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

### 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

### Problems. Universidad San Pablo - CEU. Mathematical Fundaments of Biomedical Engineering 1. Author: First Year Biomedical Engineering

Universidad San Pablo - CEU Mathematical Fundaments of Biomedical Engineering 1 Problems Author: First Year Biomedical Engineering Supervisor: Carlos Oscar S. Sorzano September 15, 013 1 Chapter 3 Lay,

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### 4.9 Markov matrices. DEFINITION 4.3 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if. (i) a ij 0 for 1 i, j n;

49 Markov matrices DEFINITION 43 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if (i) a ij 0 for 1 i, j n; (ii) a ij = 1 for 1 i n Remark: (ii) is equivalent to AJ n

### 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

### 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

### 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