MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system of linear equations for x, x 2, x 3, and x 4, the following matrix is obtained: 2 5 0 3 2 3 0 0 0 0 0 0 0 0 What is the set of solutions? 3. (0 pts (a Given a 4 5 matrix A find the elementary matrix E that corresponds to adding twice the second row to the third row. (b Find E. 4. (0 pts Multiply to find the product: 2 2 0 4 0 0 3 2 2 3 2 0 5. (5 pts Use row reduction to find the inverse of the matrix 2 3 0 0 2 0 2 6. (5 pts Suppose that A = (a ij is a k m matrix, B = (b ij a m n matrix, and C = AB (a In terms of the a ij and b ij give the formula for c ij. (b Let D = C T. In terms of the a ij and b ij give the formula for d ij. (c Let E = B T A T. In terms of the a ij and b ij give the formula for e ij. 7. (0 pts Suppose that U, V, and W are nonsingular n n matrices. find the inverse for the partitioned matrix 0 U 0 V 0 0 0 0 W 8. (0 pts Suppose that x and x 2 are two different solutions to Ax = b. (a Find a non-trivial solution to the homogeneous system Ax = 0.
(b Find an infinite set of solutions to the problem Ax = b. 9. (0 pts Find the LU decomposition for the following matrix: A-x2. (20 pts Define: (a Span(v, v 2,..., v k. (b Spanning set for the subspace S. (c Linear dependence. (d Basis for the vector space V. ( 2 5 3 2. (0 pts Let A be a nonsingular matrix. Show that det(a = /det(a. 3. (0 pts Use Cramer s rule to solve the following system: ax + by =, bx + ay = (a 2 + b 2 > 0. 4. (5 pts Let x := 2 0, x 2 := 2 3 5, x 3 := 4 7 3 5. (a Determine if these vectors are linearly dependent. (b Find a basis for the subspace spanned by these vectors. (c Find a basis for the null space of A := (x, x 2, x 3. 5. (0 pts Find the determinant of the following matrix by first row reducing to triangular form: 2 0 6 5 2 9 6 4 3 7 2 3 6. (0 pts Answer these questions for the matrix A in problem 4c: A = 2 4 7 3 3 2 0 5 5 2
(a Does Ax = ( 2, 2, 2, 4, 0 T have a solution? Explain. (b Suppose the equation Ax = b has a solution. Is it unique? Explain. 7. (5 pts In each case below determine if S is a subspace. (a S := {p P 5 p ( = 0} (b S := {x = (x, x 2, x 3, x 4 T R 4 x = x 3 3 } (c S := {M R 2 2 M T = M} 8. (0 pts Let α := {(, 2 T, (, T } and β := {(2, T, (2, T } be two ordered bases for R 2. Let [x] α = (0, 3 T, the coordinates of x with respect to the basis α. Find [x] β. A-x3. (5 pts Let A := 2 5 3 0 2 2 2 20 0. (a Find a basis for the row space of A. (b Find a basis for the column space of A. (c Find a basis for the null space of A. 2. (20 pts Let L be defined on P 3 (the vector space of polynomials of degree 2 by L(p = q where (a Find the range of L. (b Find the the kernel of L. q(x = p(x + p (( x 2 (c Find all solutions for the equation L(p = x 2. (d Find all solutions for the equation L(p =. 2 0 p(s ds. (e Find the matrix representation of L : P 3 P 3 with respect to the standard basis {, x, x 2 }. 3. (0 pts Let A be an m n matrix. Show that its null space N(A is the orthogonal complement of the range (column space of A T. 4. (30 pts Complete the following definitions: (a The vectors {x, x 2,, x k } are said to be linearly dependent if (b X Y = V means that (c The rank of a matrix is (d The dimension of a vector space is (e {x, x 2,, x n } is a spanning set for a subspace S if (f {x, x 2,, x n } is a basis for the vector space V if 3
5. (5 pts In each case below determine if S is a subspace. (a S := {p P 5 p(s ds = } (b S := {x = (x, x 2, x 3, x 4 T R 4 x 2 = x2 3 } (c S := {M = (m ij R n n n i= m ij = 0 for all j =, 2,..., n} 6. (0 pts Let A R m n and b R m. A-final (a Describe the relationship between N(A and uniqueness of solutions to Ax = b. (b Describe the relationship between N(A T and existence of a solution to Ax = b.. (0 pts Find the vector projection of (, 2, 3, 4 T on (4, 3,, 2 T. 2. (25 pts Define the following terms (in a general vector space, not necessarily R n : (a Inner product. (b Orthonormal set of vectors (c Linear independence (independently from the term linear dependence. (d Norm. (e Direct sum of two subspaces. 3. (20 pts Let L be defined on P 3 (the vector space of polynomials of degree less than 3 by L(p = q where q(x = 4p(x 3xp (x + x 2 p (x. (a Find the range of L in the form Span(.... (b Find the the kernel of L in the form Span(.... (c Find all solutions for the equation L(p = 6 x. (d Find the matrix representation of L : P 3 P 3 with respect to the ordered basis [, x, x 2 ]. 4. (5 pts Finding eigenvalues and eigenvectors: (a The matrix 0 3 2 2 0 5 8 0 has eigenvalue. Find a corresponding eigenvector. (b Find the eigenvalues of the matrix ( 29 3 4
(c Given that the matrix ( 2 3 3 2 has eigenvalues and 5 with respective eigenvectors (, T following problem: x = 2x + 3x 2, x 2 = 3x 2 2x, x (0 =, x 2 (0 = 0. Put your answer in the form x (t =..., x 2 (t =.... and (, T, solve the 5. (20 pts Let A be an n n matrix. Suppose that there exists a nontrivial solution to the homogeneous system Ax = 0. Using each of the key concepts listed below give a statement that is equivalent to this supposition. (a Rank of A. (b Uniqueness of solutions to Ax = b. (c Eigenvalues of A. (d Determinant of A. (e Rank of A T. (f Nullity of A. (g Nullity of A T. (h Row echelon form of A (careful! 6. (0 pts Let B := [M, M 2, M 3, M 4 ] be the ordered basis of R 2 2 defined by: ( ( ( ( 0 0 0 M :=, M 0 0 2 :=, M 0 3 :=, M 0 4 := With respect to this basis find the coordinate vector of ( 4 3. 2 7. (5 pts Given the vectors x = (, 4, 3 T, y = (0,, T and z = (2, 0, T, find the values of both α and β such that the vector x + αy + βz is orthogonal to both y and z. 8. (0 pts Use row reduction to find the determinant of the following matrix: 0 3 2 3 4 4 2 2 2 2 7 9. (5 pts Let x := 2, x 2 := 2 3, x 3 := and let S be the subspace of R 4 spanned by these vectors. 5 4 7 3,.
(a Find a basis for S in R 4. (b Find a basis for the subspace S spanned by the vectors (x, x 2, and x 3. 0. (0 pts Let A be an m n matrix. Note that A is not necessarily square. B-x (a Using only the concept of orthogonality, prove that for an m n matrix A we have N(A T = R(A. (b If the rows of A are linearly independent, what can you say about the solution set for the problem Ax = b? Explain by making use of the concept of rank.. (0 pts Suppose that A and C are nonsingular matrices. Find the matrices α, β, γ, δ, such that ( ( ( α β A B P Q =. γ δ 0 C 0 R You may assume that all submatrices are square and of the same dimension. 2. (0 pts Describe the elementary row operations. 3. ( 0 pts After performing row reductions on the augmented matrix for a certain system of linear equations for x, x 2, x 3, x 4 and x 5, the following matrix is obtained: 0 3 2 3 0 2 3 5 0 0 0 2 0 0 0 0 0 4 What is the set of solutions? 4. (5 pts Give the elementary matrix that will interchange the second and fourth rows of 6 n matrix. 5. (5 pts Evaluate: 0 0 4 0 0 2 0 T 0 0 0 0 ( 4 + 2 6 2 6. (5 pts Give the precise definition of the inverse of a matrix A. 7. (0 pts Precisely state a theorem that gives two statements that are equivalent to nonsingularity of a square matrix. 8. (5 pts Use row reduction to find the inverse of the matrix 2 0 0 2 6
9. (0 pts Suppose that A and B are two nonsingular n n matrices. Prove that their product, AB is also nonsingular. 0. (0 pts Suppose that Ax = 0 has a nontrivial solution. Prove that that if the system Ax = b has one solution, then it has infinitely many solutions. B-x2. (25 pts Define: (a Linear independence. (b Subspace. (c Dimension of a vector space. (d Determinant of a square matrix. (e Span{x, x 2,, x k } 2. (20 pts Let A = (a ij be a nonsingular n n matrix and let B = (b ij = A. (a Define the cofactor, A ij corresponding to the entry a ij. (b Using the cofactors, give a formula for b ij. (c Prove that det(b = /det(a. (d Let C be an n n matrix such that AC = I. Prove that C = A. 3. (5 pts Let A = After row reduction this matrix becomes B := 8 8 5 2 2 0 0 6 3 0 0 5 6 6 4 2 4 3 0 0 (a Find a basis for the row space of A. (b Find a basis for the column space of A. (c Find a basis for the null space of A. 9 0 9 0 4 7 0 9 0 0 34 4 0 0 0 9 4 0 0 0 0 0 0 0 4. (0 Answer these questions for the matrix A in problem 3: (a Does Ax = b have a solution for all b R 6? Explain... (b Suppose the equation Ax = b has a solution. Is it unique? Explain. 5. (0 pts Let X = (x, x 2,, x n be n linearly independent vectors in R n and let c = (c, c 2,, c n. Show that y = c x + c 2 x 2 + + c n x n if and only if c = X y. 6. (5 pts Let α := {(2, T, (, T } and β := {(, 2 T, ( 3, T } be two ordered bases for R 2. Let [x] α = (, 2 T, the coordinates of x with respect to the basis α. Find [x] β. 7
7. (5 pts An n n matrix A is said to be orthogonal if A = A T. Prove that an orthogonal matrix has determinant equal to or equal to -. State the theorem(s that you use. B-x3. (0 pts Let A be an m n matrix. Show that its null space N(A is the orthogonal complement of the range (column space of A T. 2. (0 pts Find the vector projection of (, 0, 2, 2 T on (3,,, 5 T. 3. (35 pts Complete the following definitions: (a The vectors {x, x 2,, x k } are said to be linearly dependent if (b The vector space V is said to be the direct sum of the two subspaces X and Y if complementary subspaces if (c The rank of a matrix is (d The dimension of a vector space is (e {x, x 2,, x n } is a spanning set for a subspace S if (f {x, x 2,, x n } is a basis for the vector space V if (g Z is said to be the direct sum of the subspaces X and Y if 4. (20 pts Let P 5 be the vector space of polynomials of degree less than 5 and let L : P 5 P 5 be defined by L(p = q where q(x = 3p(x 3xp (x + x 2 p (x. (a Find the range of L in the form Span(.... (b Find the the kernel of L in the form Span(.... (c Find all solutions for the equation L(p = 9 4x. (d Find the matrix representation of L with respect to the ordered basis [, x, x 2, x 3, x 4 ]. 5. (0 pts Given the vectors x = (, 4, 3 T and y = (0,, T, find the value of α such that the vector x + αy is orthogonal to y. 6. (5 pts Let B-final x := 2 0, x 2 := 2 3, x 3 := and let S be the subspace of R 4 spanned by these vectors. (a Find a basis for S. 0 2, (b Find a basis for the subspace S spanned by the vectors x, x 2, and x 3.. (0 pts Suppose we know the following facts about the real 4 4 matrix A: 8
+ i is an eigenvalue of A, det(a = 40, tr(a =. Find the other three eigenvalues of A 2. (5 pts Let α := [(3, 3 T, (, T ] and β := [(, 2 T, (3, T ] be two ordered bases for R 2. Let [x] α = (3, 2 T, the coordinates of x with respect to the basis α. Find [x] β, the coordinates of x with respect to the basis β. 3. (5 pts Finding eigenvalues and eigenvectors: (a The matrix 4 2 2 3 8 5 8 4 has eigenvalue 3. Find a corresponding eigenvector. (b Find the eigenvalues of the matrix (c Solve the system of differential equations ( 2 5 y = 3y + 2y 2 y 2 = 3y 2y 2 You may use the fact that the coefficient matrix has eigenvalues 4 and 3 with respective eigenvectors (2, T and (, 3 T. 4. (25 pts Let A be an n n matrix. Suppose that there exists a nontrivial solution to the homogeneous system Ax = 0. Name 8 statements that are equivalent to this supposition and that involve the given key concepts: (a Rank of A. (b Uniqueness of solutions to Ax = b. (c Eigenvalues of A. (d Determinant of A. (e Rank of A T. (f Nullity of A. (g Nullity of A T. (h Row echelon form of A (careful! 5. (25 pts Let B := [M, M 2, M 3, M 4 ] be the ordered basis of R 2 2 defined by: ( ( ( ( 0 0 0 0 0 0 M :=, M 0 0 2 :=, M 0 0 3 :=, M 0 4 := 0 Define the linear transformation L := A (A + A T.. 9
(( a b (a What is L c d? (b Find the range of L in the form Span(. (c Find the kernel of L in the form Span(. (d Find the matrix representation of L with respect to the basis B (e Find all the solutions to the problem 6. (0 pts L(A = ( 4 0 0 6 (a State what it means for an n n matrix to be diagonalizable. (b State a criterion that ensures that a square matrix is diagonalizable. (c Prove that if A and B are similar square matrices then they have the same characteristic polynomial. 7. (25 pts Define the following terms (in a general vector space, not necessarily R n : (a Inner product. (b Linear transformation on the vector space V. (c Linear independence (independently from the term linear dependence. (d Norm. (e Direct sum of two subspaces. 8. (0 pts Use row reduction to find the determinant of the following matrix: 0 3 2 3 4 4 2 2 2 2 7 9. (0 pts Let A be an m n matrix. Note that A is not necessarily square. (a If the columns of A are linearly independent, what can you say about the solution set for the problem Ax = b? (b If the rows of A are linearly independent, what can you say about the solution set for the problem Ax = b? 0. (5 pts Let A be a nonsingular matrix. Prove that for any positive integer m, A m is also nonsingular. Hint: What is the inverse of A m?. (0 pts - Bonus Let A = (a ij be an n n matrix. Let a j denote its i th column and let A ij denote its cofactors. Let b denote the column (A n, A 2n,, A nn T of cofactors. (a Show that det(a = b a n. (b Show that b a i for i =, 2,, n. Hint: Last column of A. When n = 3, then b is simply the cross product of a and a 2. 0
C-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system of linear equations for x, x 2, x 3, and x 4, the following matrix is obtained: 2 5 0 3 2 3 3 0 0 0 0 0 0 0 0 0 What is the set of solutions? 3. (0 pts (a Given a 4 5 matrix A find the elementary matrix E that corresponds to multiplying the third row by 2. (b Find E. 4. (0 pts Multiply to find the product: 3 2 2 3 2 0 ( 2 0 4 0 0 5. (5 pts Use row reduction to find the inverse of the matrix 2 2 3 2 0 2 6. (0 pts Suppose that A = (a ij is a k m matrix, B = (b ij a m n matrix, and C = AB (a In terms of the a ij and b ij give the formula for c ij. (b Let D = C T. In terms of the a ij and b ij give the formula for d ij. (c Let E = FG where F = B T and G = A T. In terms of the a ij and b ij and without using the fact that B T A T = (AB T, give the formula for e ij. 7. (5 pts Suppose that A is an n n matrix having identical first and second columns. Without using determinants, show that A is singular. 8. (0 pts Suppose that x and x 2 are two different solutions to Ax = b. (a Find a non-trivial solution to the homogeneous system Ax = 0. (b Find an infinite set of solutions to the problem Ax = b. 9. (0 pts Suppose that B is a singular n n matrix and that K is any other n n matrix. Prove that the product BK is also singular.
0. (5 pts Suppose that A = (p, q, r where p = (, 3, 5, 6, 7 T, q = ( 7, 2,, 3, 5 T and r = ( 5, 6, 7, 0, 2 T. Let b = p + 2q + 4r. Find a solution to the equation Ax = b.. (5 pts Explain why the following is not true and give the correct identity: C-x2. (20 pts Define: (a Spanning set for the subspace S. (b Linear independence. (c Basis for the vector space V. (d Subspace. (A B 2 = A 2 2AB + B 2. 2. (0 pts Two n n matrices A and B are said to be similar if there is a nonsingular matrix S such that B = S AS. Prove that two similar matrices have the same determinant. State the theorem(s that you use. 3. (0 pts Let A be a 7 5 matrix whose rank is 3. (a What is the dimension of the nullspace of A? (b What is the dimension of the range of A? (c What is the dimension of the nullspace of A T? 4. (5 pts Let α := [(3, T, (, T ] and β := [(4, T, (4, 2 T ] be two ordered bases for R 2. Let [x] α = (7, 3 T, the coordinates of x with respect to the basis α. Find [x] β. 5. (5 pts Let A := 2 5 3 4 0 0 4 0 0 0 5. (a Find a basis for the row space of A. (b Find a basis for the column space of A. (c Find a basis for the null space of A. 6. (0 Answer these questions for the matrix A in the previous problem: (a Does Ax = (4, 6, 2, 0 T have a solution? Explain. (b Suppose the equation Ax = b has a solution. Is it unique? Explain. 7. (0 pts Let {x, x 2,, x n } be n linearly independent vectors in R n and let c = (c, c 2,, c n T. Define the n n matrix: X := (x, x 2,, x n. Show that y = c x + c 2 x 2 + + c n x n if and only if c = X y. 2
8. (20 pts Let {x, x 2,, x k } and {y, y 2,, y m } be two linearly independent sets of vectors in a vector space V. Let X := Span{x, x 2,, x k } C-x3 Prove the following: Y := Span{y, y 2,, y m }. (a If the set of vectors {x, x 2,, x k, y, y 2,, y m } is linearly independent then X Y = {0}. (b If the set of vectors {x, x 2,, x k, y, y 2,, y m } is linearly dependent then X Y contains a nonzero vector.. (20 pts Let L be defined on P 4 (the vector space of polynomials of degree 3 by L(p = q where q(x = 2 p (0x 2p(x + xp (x. (a Find the range of L. (b Find the the kernel of L. (c Find all solutions for the equation L(p = x 2 +. (d Find the matrix representation of L : P 4 {, x, x 2, x 3 }. P 4 with respect to the standard basis 2. (0 pts Let A R m n and b R m. (a Describe the relationship between N(A and uniqueness of solutions to Ax = b. (b Describe the relationship between N(A T and existence of a solution to Ax = b. 3. (5 pts Given the vectors x = (,, T, y = (0,, T and z = (,, T, find the values of both α and β so that the vector x + αy + βz is orthogonal to both y and z. 4. (30 pts Define the following terms (in a general vector space, not necessarily R n : (a Inner product. (b Linear transformation. (c Orthogonal subspaces. (d Norm. (e Direct sum of two subspaces. (f Kernel of a linear transformation. 5. (0 pts Find the vector projection of (,,,, 3 T on (3, 2, 2, 2, 2 T. 3
6. (5 pts Let C-final x := 3, x 2 := 2 0, x 3 := and let S be the subspace of R 4 spanned by these vectors. (a Find a basis for S. 4 7 6 7, (b Find a basis for the subspace S spanned by the vectors (x, x 2, and x 3.. (20 pts Let A = 9 0 25 4 7 4 0 0 8 3 24 2 7 0 8 8 9 3 0 3 33 5 6 0 27 8 22 After row reduction this matrix becomes 0 0 2 2 0 0 2 3 B := 0 0 0 6 0 0 0 0 4. 0 0 0 0 0 0 (a Find a basis for the row space of A. (b Find a basis for the column space R(A, of A. (c Find a basis for the null space, N(A. (d Find a basis for N(A. 2. (5 pts Let A R n n and x R n and suppose that Ax and y are orthogonal. Show that A T y must also be orthogonal to x. Each step must be carefully justified! 3. (0 pts Let A be an m n matrix. Note that A is not necessarily square. (a If the columns of A are linearly independent, what can you say about the solution set for the problem Ax = b? (b If the rows of A are linearly independent, what can you say about the solution set for the problem Ax = b? 4. (5 pts Suppose. A := 27 6 2 28 6 36 8 7 = S QS 2 9 2 20 5 3 52 0 0 0 2 0 0 0 8 5 4 9 2 0 Find A 5. 4
5. (5 pts Suppose we know the following facts about the real 4 4 matrix A: 2 + i is an eigenvalue of A, det(a = 60, tr(a = 3. Find the other three eigenvalues of A 6. (24 pts Let A be an m n matrix, not necessarily square. Suppose that for each b R m there exists a solution x to the system Ax = b. Being as specific and precise as you can, answer the following. If the given information is not enough to answer the question then write not enough information. (a Rank of A = (b Solutions to Ax = b are unique (yes or no. (c Rows of A are linearly independent (yes or no. (d Columns of A are linearly independent (yes or no. (e Rank of A T = (f Nullity of A = (g Nullity of A T = (h Number of leading ones in the row echelon form of A = (i Number of free variables for the system Ax = b = (j Relative size of m and n: Is m n or n m? (k Dimension of the orthogonal complement of N(A = (l Dimension of the orthogonal complement of N(A T = 7. (0 pts Find the determinant of the following matrix by first row reducing to triangular form: 6 4 5 4 0 2 3 0 2 5 3 4 4 2 5 8. (5 pts Let x := 2, x 2 := 2 3 5, x 3 := In order to minimize the amount of work, read the entire problem and determine a strategy - you may not want to do the problem in the order in which they are stated! (a Determine if these vectors are linearly dependent. (b Find a basis for the subspace spanned by these vectors. (c Find a basis for (Span(x, x 2, x 3. 9 7 2 5. 5
9. (20 pts Let B := [, e x, e x, sin(x, cos(x] be the ordered basis for a subspace S of the vector space C[0, 2π]. Define the linear transformation L := f(x f(0 + f(x f (x. (a Find the matrix A that represents L with respect to the basis B (b Find a basis the Null space of A. (c Find the kernel of L. (d Find all the solutions to L(f = 4. 0. (30 pts Complete the following definitions: (a The vectors {x, x 2,, x k } are said to be linearly dependent if... (b The rank of a matrix is... (c The dimension of a vector space is... (d {x, x 2,, x n } is a spanning set for a subspace S if... (e {x, x 2,, x n } is a basis for the vector space V if... (f Let V be a vector space and {x, x 2,, x k, } V. Then the span Span{x, x 2,, x k, } is.... (0 pts Let A R m n and b R n. (a Describe the relationship between N(A and uniqueness of solutions to Ax = b. (b Describe the relationship between N(A T and existence of a solution to Ax = b. 2. (5 pts Finding eigenvalues and eigenvectors: (a The matrix 2 2 2 has eigenvalue 4. Find a corresponding eigenvector. (b Find the other eigenvalue(s of the above matrix. (c Find the eigenvalues of the matrix ( 3 2 5 3. (5 pts Suppose that two n n matrices A and B have the same n distinct eigenvalues. Show that A and B are similar (i.e. There exist an n n matrix S such that SAS = B. 4. (0 pts Let g i be the i th column of a n n matrix G and let h T i be the i th row of the matrix G. Let X be the subspace spanned by {g, g 2,, g k } and let Y be the subspace spanned by {h k+, h k+,, h n }. Show that X and Y are orthogonal complements in R n. Hint G G = I. D-x. (0 pts Name the three elementary row operations. 6
2. ( 20 pts After performing row reductions on the augmented matrix for a certain system of linear equations for x, x 2, x 3, and x 4, the following matrix is obtained: What is the set of solutions? 2 5 0 0 2 3 0 0 0 0 0 0 0 0 3. (5 pts Prove that if A and B are two nonsingular n n matrices then (AB = B A 4. (0 pts Suppose that A and B are two given n n matrices and that A has an inverse. Solve the following equation for the unknown n n matrix X:. 5. (20 pts Consider the following system: AXA + A 2 = B x + y + z = 0 2x y 2z = 0 x + 4y + 5z = 0 (a Find the determinant of the coefficient matrix A. (b State an appropriate theorem and give a brief argument to show that the above system has a nontrivial solution. 6. (0 pts Let A be a 5 7 matrix. (a Find the elementary matrix E such that EA is the matrix A with rows 2 and 4 interchanged. (b Find E. 7. (5 pts Let A and B be two n n matrices and suppose that B is singular. D-x2 (a Show that the n n matrix AB is also singular. (b Show that the matrix BA is also singular (Hint: (BA T =.. (5 pts Let A = 0 3 5 7 3 5 7 8 9 9 3 5 7 2 7 2 3 9 5 24 33 45 2 3 5 7 0 2 5 8 2 6 7. 7
After row reduction this matrix becomes 2 3 4 5 6 0 2 3 4 5 B := 0 0 0 2 3 0 0 0 0 0. 0 0 0 0 0 0 0 0 0 0 0 0 (a Find a basis for the null space of A. (b Find a basis for the row space of A. (c Find a basis for the column space of A. 2. (0 pts Answer these questions for the matrix A in the previous problem: (a Does Ax = b have a solution for all b R 6? Explain. (b Suppose the equation Ax = b has a solution. Is it unique? Explain. 3. (5 pts Use Cramer s rule to find x 2. Evaluate determinants by using row operations: 4. (25 pts Define: (a Linear independence. (b Subspace. (c Basis. x + x 2 = 0 x 2 + x 3 2x 4 = x + 2x 3 + x 4 = 0 x + x 2 + x 4 = 0 (d Spanning set for a subspace W of a vector space V. (e Span{x, x 2,, x k }. 5. (0 pts Suppose that {v, v 2,, v n } is a set of linearly independent vectors in a vector space V. Prove that if v n V but v n / span(v, v 2,, v n then {v, v 2,, v n } is a linearly independent set of vectors. 6. (5 pts Let and E := {(, 3 T, (, T } F := {(3, 0 T, (2, T } be two ordered bases for R 2. Let [x] E = (, 5 T, be the coordinates of x with respect to the basis E. Find [x] F. 8
7. (0 pts Let B := [M, M 2, M 3, M 4 ] be the ordered basis of R 2 2 defined by: ( ( ( /2 /2 /2 /2 /2 /2 M :=, M /2 /2 2 :=, M /2 /2 3 := /2 /2, M 4 := ( /2 /2 /2 /2. D-x3 Find the coordinate vector of the matrix with respect to the basis B. X := ( 4 0 2 6. (0 pts a The matrix has eigenvalue. (a Find a corresponding eigenvector. (b Find the eigenvalues of the matrix 2 3 2 2 2 5 8 2 ( 4 2 2. (20 pts Give a precise statement of each: (a The Matrix Representation Theorem for linear transformations. (b A theorem about diagonalizability. 3. (0 pts In the space C[0, ] with inner product f, g := 0 f(xg(x dx, find the vector projection of the function + x onto the function x 2. 4. (0 pts Find the least-squares solution to the equation Ax = b if 2 0 A := 3 and b :=. 2 5. (5 pts Suppose S = Span {(,,, 0 T, (, 0,, 0 T }. (a Find a basis for S. (b Find a basis for ( S 6. (0 pts Given the vectors (4, 4, 4, 4 T, and (6, 2, 6, 2 T and (0, 0, 4, 6 T 9
the Gram-Schmidt process is carried out. The first two orthonormal vectors obtained are Find the third orthonormal vector. 7. (30 pts Complete the following definitions: (/2, /2, /2, /2 T and (/2, /2, /2, /2 T. (a An inner product on a vector space V is (b V is the direct sum of the subspaces X and Y if (c The vectors {x, x 2,, x k } are said to be linearly dependent if (d The dimension of a vector space is (e {x, x 2,, x n } is a spanning set for a subspace S if 8. (20 pts Let A be an m n (non-square matrix. Suppose that there exist b such that Ax = b does not have a solution, but whenever a solution exists, it is unique. Complete the following statements: (a Rank of A = (b A T x = c has a solution for all c R n. This statement is (true or false? (c The bigger of m and n is (d Nullity of A= (e Nullity of A T = (f Row echelon form of A has (g Row echelon form of A T has (h A T A is a... matrix. 9. (25 pts Let B := [M, M 2, M 3, M 4 ] be the ordered basis of R 2 2 defined by: M := ( 0 0 0 ( 0, M 2 := 0 0 ( 0 0, M 3 := 0 ( 0 0, M 4 := 0. Define the linear transformation ( a b R : c d ( b d a c, and (( a b (a What is L c d? (b Find the kernel of L as span(. L := A (R(A A. (c Find the matrix representation of L with respect to the basis B (d Find the range of L as span(. 20
E-x. ( 0 pts After performing row reductions on the augmented matrix for a certain system of linear equations for x, x 2, x 3, and x 4, the following matrix is obtained: 2 7 0 0 3 3 0 0 0 2 0 0 0 0 0 What is the set of solutions? 2. (0 pts (a Given a 4 6 matrix A find the elementary matrix E that corresponds to adding twice the second row to the fourth row. (b Find E. 3. (5 pts Use row reduction to find the inverse of the matrix 0 2 2 0 2 0 5 4. (0 pts Suppose that x and x 2 are two different solutions to Ax = b. Show that det(a = 0. Explain which theorems you are using. 5. (5 pts Using row reduction, compute the determinant of the matrix 5 2 3 4 5 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 6. (5 pts Suppose that A and C are nonsingular matrices. (a In terms of A and C, find the inverse of the matrix ( A O O C (b Use part (a to find the inverse of 2 0 0 3 4 0 0 0 0 2 0 0 3 4 7. (5 pts Give the precise definition of the inverse of a matrix A. 8. (0 pts Precisely state the Equivalence of Nonsingularity theorem. 2
9. (0 pts Suppose that A and B are two nonsingular n n matrices. Prove that their product, AB is also nonsingular. E-x2. (20 pts Let A := The reduced row echelon form of A is U := (a Find a basis for the row space of A. (b Find a basis for the column space of A. (c Find a basis for the null space of A. 2 2 2 3 0 0 4 0 6 0 4 0 0 0 0 2. (0 Answer these questions for the matrix A in problem : (a Without solving the equation Ax = (2, 2, 2 T, explain why it must have a solution... (b Suppose the equation Ax = b has a solution. Is it unique? Explain. 3. (20 pts Let α := {(, 2 T, (, T } and β := {(3, T, (0, 3 T } be two ordered bases for R 2. Let [x] α = (5, 3 T, the coordinates of x with respect to the basis α. Find [x] β. 4. (20 pts Define: (a The span of a set of vectors {v, v 2,, v n }. (b Spanning set for the subspace V. (c Linear independence. (d kernel of a linear transformation L : V W. 5. (0 pts Suppose that A is a nonsingular n n matrix and that x, x 2,, x n are vectors in R n such that the set {Ax, Ax 2,, Ax n } is linearly dependent. Using only the definition of linear dependence prove that the set {x, x 2,, x n } is also linearly dependent 6. (20 pts Let L : P 4 P 4 (the vector space of polynomials of degree less than 4 by L(p = q where q(x = 4p(x 2xp (x. (a Find the matrix representation of L : P 4 S := [, x, x 2, x 3 ]. (b Find a basis for the kernel of L. (c Find a basis for the image of L. P 4 with respect to the standard basis E-final 22
. (0 pts Find the vector projection of (2, 2, 3, T on (2, 2, 0, 4 T. 2. (0 pts Let A := ( 2 a 3 5 For what value(s of a does the system Ax = 0 a nontrivial solution? 3. (20 pts Let L : R 2 2 R 2 2 (a linear transformation on the vector space of 2 2 real matrices be defined by. ( a a L : 2 a 2 a 22 ( a a 2 a 2 a 22 2a 2 a 22 a (a Find the kernel of L in the form Span(.... (b Find the matrix representation with respect to the standard ordered basis B := [M, M 2, M 3, M 4 ], where ( ( ( ( 0 0 0 0 0 0 M =, M 0 0 2 =, M 0 0 3 =, M 0 4 =, 0 4. (0 pts In P 3, the vector space of polynomials of degree less than 3, find the coordinate vector of p := x 2 + 2x + with respect to the ordered basis C := [x 2 + x, x +, ]. 5. (20 pts Finding eigenvalues and eigenvectors: (a The matrix 3 3 2 2 3 5 8 3 has eigenvalue 2. Find a corresponding eigenvector. (b Find the eigenvalues and corresponding eigenvectors for the matrix ( 3 2 5 6. (20 pts Let x = (, 2, 0 T, x 2 = (3, 2, 8 T, x 3 = (, 6, 8 T, and let S := Span(x, x 2, x 3. (a Find a basis for S (b Find a basis for S. 7. (30 pts Complete the following definitions: (a The vectors {x, x 2,, x k } are said to be linearly independent if (b Two subspaces X and Y of a vector space V are said to be orthogonal subspaces if (c The rank of a matrix is (d The set {v, v 2,, v k } is said to be a spanning set for the vector space S if (e The set {v, v 2,, v k } is said to be a basis for the vector space S if (f Let V be a vector space, S V. We say that S is a subspace of V if 23
8. (5 pts Some very short proofs. Let A be a square matrix, possibly complex. (a Show that if λ is an eigenvalue of A then it is also an eigenvalue of A T. (b Show that if λ is an eigenvalue of A then λ is an eigenvalue of A. (c Let B := A + A. Show that B is diagonalizable. 9. (5 pts Using the method of least squares find the least squares fit for a line y = αx + β through the points (, 2, (2, 6, (3, 4. 0. (5 pts Use row reduction to find the inverse of the matrix 0 2 2 0 0 0 2 5. (0 pts State 2 of the following 3 theorems: the Fundamental Subspace Theorem, the Equivalence of Nonsingularity Theorem, the Rank-Nullity Theorem. 2. (25 pts Let A be an 3 4 matrix. Suppose that all solutions of the homogeneous system Ax = 0 is of the form α(,,, T for some scalar α. Being as specific and precise as you can, answer the following. If the given information is not enough to answer the question then write not enough information. (a What is the rank of A? (b Rows of A are linearly independent (true or false. (c Columns of A are linearly independent (true or false. (d What is the nullity of A? (e What is the nullity of A T? (f What is the number of leading ones in the row echelon form of A? (g What is the number of free variables for the system Ax = b? (h What is the number of free variables for the system A T y = c? (i What is the number of zero rows in the row echelon form ofa T? (j Does the system A T y = (2, 3,, 2 T have a solution? (k What is the dimension of the orthogonal complement of N(A T? (l A must be either singular or nonsingular,(true or false 24