No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.

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1 Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points Score Total: 26

2 . Fill in the blanks in the following definitions: (I) (0 points) Let V be a vector space with basis X = (v,..., v n ) and W be a vector space with basis Y = (w,..., w m ). Let F be a linear transformation from V to W. Then the matrix of F with respect to the bases X and Y, Y F X, is the m n matrix [fill in the matrix below] whose i-th column is the coordinate vector of with respect to the basis. 2. Fill in the blanks in the following statements from the text. (I) (6 points) Given a vector v and a subspace W of R n, the orthogonal projection w of v onto W is unique, and has the property that if w is any vector in W with w w then < (II) (9 points) If A is an m n matrix, regarded as a linear transformation from R n to R m, then the rank r of A and the dimension k of Null(A) are related by the equation: = + 3. Indicate whether each statement is true or false by circling the appropriate answer. (I) (5 points) [ True False ] Define two subspaces V and W of R 3 by V = Span(v, v 2 ) and W = Span(w, w 2 ), where v = (, 3, 2), v 2 = (0, 5, 0), w = (, 0, 0), w 2 = (0, 3, 2). Then V and W are different subspaces of R 3. (II) (5 points) [ True False ] If A is an n n matrix whose row space and column space are equal, as subspaces of R n, then A is a symmetric matrix. (III) (5 points) [ True False ] If A is an n n matrix which is diagonalizable, then A has n distinct eigenvalues. (IV) (5 points) [ True False ] If I : R n R n is the identity map, and X and Y are any two bases of R n, then the matrix Y I X (of I with respect to X and Y ) is the identity matrix. (V) (5 points) [ True False ] If 0 is the only eigenvalue of a square matrix A, then A is the zero matrix. (VI) (5 points) [ True False ] The transpose of an elementary matrix is an elementary matrix. Page 2

3 (VII) (5 points) [ True False ] The norm is additive: if x, y R n are vectors then x + y = x + y (VIII) (5 points) [ True False ] Adding 2 to every number in the third row of a 4 4 matrix is an elementary row operation on that matrix. (IX) (5 points) [ True False ] If {v, v 2,..., v n } is a set of n orthonormal vectors in R n, then the matrix whose columns are v, v 2,..., v n has determinant. (X) (5 points) [ True False ] Suppose that A is a symmetric, n n matrix. Then there is an orthonormal basis of R n consisting of eigenvectors of A. 4. Fill in the blanks in the following result from the text: If X is a subset of a vector space V, then X is a vector space if it satisfies the following 3 properties: (I) (2 points) 0 (II) (4 points) If u, v, then (III) (4 points) If c R and v, then. 5. Let V = Span(p, p 2, p 3, p 4 ) be the subspace of P 2 defined by the following polynomials: p (x) = + 2x x 2 p 2 (x) = 5 + 4x x 2 p 3 (x) = 3 + x 2 p 4 (x) = 2 + 2x 2x 2 Recall that P 2 is the space of polynomials of degree less than or equal to 2. (I) (6 points) Find, with proof, the dimension of V. Page 3

4 (II) (4 points) Find a basis of V. 6. Let L: R 2 R 2 be a linear transformation. With respect to the basis X = {v, v 2 }, where [ ] [ ] 3 v = v 2 2 = 4 The matrix of L is given by XL X = [ ] In what follows, we write I : R 2 R 2 for the identity map of R 2 to itself, and E for the standard basis {e, e 2 } of R 2. (I) (5 points) Find the matrix E I X. Page 4

5 (II) (5 points) Find the matrix X I E. (III) (8 points) Find the matrix E L E. 7. Let v R n be any vector and let {w,..., w m } be an orthogonal basis of a subspace W of R n. (I) (5 points) Write the formula for the orthogonal projection proj W (v) of v onto W. To get credit, you must use the variables defined above. Copying the variable names from the statement given in the text will not get credit. Hint: Your formula should start proj W (v) =. Page 5

6 (II) (5 points) [ True False ] The formula above for the orthogonal projection holds when {w,..., w m } is any basis of W. 8. (2 points) Let X = {v, v 2, v 3, v 4 }, where 0 v = 2 v 2 = 0 v 3 = v 4 = 0 0 You may assume in this problem that X is a basis for R 4. Apply the Gram-Schmidt process to X to find an orthogonal basis Y of R Define the vectors v R 3 and the matrix A by 4 v = 3 A = (I) (4 points) Find Av. Page 6

7 (II) (6 points) What does this say about eigenvalues and eigenvectors of A? (III) (0 points) Find all eigenvalues of A. 0. The matrix B below has characteristic polynomial (λ 2) 2 (λ ). (You may assume the given characteristic polynomial is correct and are not required to verify it.) 0 B = (I) (2 points) What are the eigenvalues of B, and what multiplicity does each of these eigenvalues have? Page 7

8 (II) (0 points) For each eigenvalue of B, find an a basis of the associated eigenspace.. For each of the following matrices, say whether or not it has a square root. (A matrix A has a square root if there is a matrix B such that B 2 = A.) If the matrix does not have a square root, explain why it doesn t. If it does have a square root, explain how you know that. 0 0 (I) (7 points) A = Page 8

9 (II) (7 points) A = [ ] For the following multiple choice questions, you may give one (and only one) answer. Select the single best option! (I) (5 points) Let V be a vector space, and let X be a subset of V. What does it mean when we say that X is linearly independent? (a) All the elements of X are distinct from each other. (b) X is closed under both addition and scalar multiplication. (c) X is a basis of V. (d) The number of elements of X is less than or equal to the dimension of V. (e) Every element in V is a linear combination of elements in X. (f) The only way to write 0 as a linear combination of elements of X is the zero combination (where one takes zero multiples of each element of X). (I) (II) (5 points) Which of the following functions T : R 3 R 3 is not a linear transformation? (a) T (x, y, z) = (x, 2y, 3x y) (b) T (x, y, z) = (x y, 0, y z) (c) T (x, y, z) = (0, 0, 0) (d) T (x, y, z) = (x,, z) (e) T (x, y, z) = (2x, 3y, 5z) (II) (III) (5 points) Which of the following statements is not true of all linear transformations T : R 4 R 4? (a) There is a 4 4 matrix A such that T (x) = A x for all x R 4. (b) T ( x) = T (x) for all x R 4. (c) T (x y) = T (x) T (y) for all x, y R 4. (d) T (0) = 0, where 0 R 4 is the zero vector. (e) T (7 x) = 7T (x) for all x R 4. (III) Page 9

10 (IV) (5 points) Suppose A is an m n matrix, regarded as a linear transformation from R n to R m. Which of the following statements is not always true? (a) The null space of A is a subspace of R n. (b) The image of A is a subspace of R m. (c) The image of A is R m if and only if the null space of A is zero. (d) The dimension of the image of A is less than or equal to n. (e) The dimension of the image of A is less than or equal to m. (IV) (V) (5 points) Let X = {x, x 2, x 3, x 4 } be a set of 4 linearly independent vectors in R 6. Let A be a 6 6 matrix of rank 5. Then the dimension of Span(A x, A x 2, A x 3, A x 4 ) (a) is equal to 4 since 5 > 4. (b) must be either 3 or 4. (c) must be, 2, or 3. (d) is equal to 5, since A has rank 5. (e) can be any number between 0 and 4 inclusive. (V) (VI) (5 points) Suppose A is a m n matrix. Which of the following is equal to the rank of A? (a) The dimension of V, where V the column space of A T. (b) The dimension of the null space of A T. (c) The number of free variables in the system of equations A x = 0. (d) The dimension of V, where V is the null space of A. (e) None of the above. (VI) (VII) (5 points) Suppose that A and B are n n matrices and A is similar to B, i.e. there is an invertible n n matrix P such that P AP = B. Which of the following statements is most accurate? (a) A 2 is always similar to B 2. (b) A 2 need not be similar to B 2. (c) A 2 is never similar to B 2. (d) A 2 is similar to B 2 if and only if A and B are diagonalizable. (e) A 2 is similar to B 2 if and only if det(a 2 ) = det(a). (VII) Page 0

11 (VIII) (5 points) If A is an m n matrix, regarded as a linear transformation from R n to R m, and we change A by doing row operations, then (VIII) (a) The image, kernel, and dimension of the kernel may change, but the rank does not change. (b) The image, kernel, rank, and dimension of the kernel may all change. (c) The image, rank, and kernel may change, but the dimension of the kernel does not change. (d) The image and kernel may change, but the rank and dimension of the kernel do not change. (e) The image may change, but the kernel, rank, and dimension of the kernel do not change. (f) The image, kernel, rank, and dimension of the kernel all do not change. (IX) (5 points) Let P 3 be the vector space of all polynomials of degree less than or equal to 3. Write D for the derivative, regarded as a linear transformation from P 3 to P 2. Then (a) The dimension of the kernel of D is 2. (b) The polynomial 2x + is in the kernel of D. (c) The polynomial 2x 3 + is in the image of D. (d) The kernel of D consists of all polynomials with zero constant term. (e) The rank of the matrix of D is 3. (IX) (X) (5 points) The dimension of the null space of a 3 5 matrix (a) Can be any number from two to five. (b) Can be any number from zero to two. (c) Must be two. (d) Must be three. (e) Can be any number from zero to three. (f) Must be zero. (g) Can be any number from zero to five. (X) (XI) (5 points) The rank of a 5 3 matrix (a) Can be any number from zero to two. (b) Must be three. (c) Can be any number from two to five. (d) Must be zero. (XI) Page

12 (e) Can be any number from zero to three. (f) Must be two. (g) Can be any number from zero to five. (XII) (5 points) Let A be a 5 5 matrix with determinant 6, and let B be a 5 5 matrix with determinant 4. What is the determinant of A + B? (a) /4. (b) There is insufficient information to answer the question. (c) /6. (d) 0. (e) 0. (f) 24. (g). (XII) (XIII) (5 points) Let A be a 4 4 matrix with determinant 3. What is the determinant of 2A? (a) 48. (b) 3. (c) 6. (d) There is insufficient information to answer the question. (e) 2. (f) 24. (g) 96,608. (XIII) (XIV) (5 points) Let A be a 4 4 matrix with determinant 3. Let B be the matrix formed from A by subtracting two times the third row from the first row. What is det B? (a) -9. (b) 0. (c) 3. (d) -3. (e) 6. (f) -6. (g) There is insufficient information to answer the question. (XIV) Page 2

13 (XV) (5 points) If one replaces a matrix with its transpose, then (XV) (a) The kernel may change, but the image, rank, and dimension of the null space do not change. (b) The image, kernel, rank, and dimension of the null space may all change. (c) The image, kernel, and dimension of the null space may change, but the rank does not change. (d) The image, kernel, rank, and dimension of the null space all do not change. (e) The image, rank, and kernel may change, but the dimension of the null space does not change. (f) The image may change, but the kernel, rank, and dimension of the null space do not change. (g) The image and kernel may change, but the rank and dimension of the null space do not change. (XVI) (5 points) Let A be an invertible 5 5 matrix, which can be regarded as a linear transformation from R 5 to R 5. Which of the following statements is not necessarily true? (a) The determinant of A is non-zero and the kernel of A is {0}. (b) The rows of A form a basis for R 5. (c) There are five distinct eigenvalues. (d) All the eigenvalues of A are non-zero. (e) The columns of A form a basis for R 5. (f) The image of A is R 5. (XVI) Page 3