Math 240, Second Exam Review Questions 2022

Math 240, Second Exam REVIEW QUESTIONS 1. Short Answer: (a) Finish the definition: The set of vectors {v1 , . . . , vk } spans set V if: (b) Finish the definition: The set of vectors {v1 , . . . , vk } for a basis for vector space V if: (c) Finish the definition: The rank of a matrix is: (d) How was the matrix-matrix product AB defined? (e) Finish the definition: The n × n matrix A is invertible if: (Note that this is the definition, not something equivalent to the definition). (f) If A is an m × n matrix, the column space of A is a subspace of IR? , and it is defined as: (g) If A is an m × n matrix, the null space of A is a subspace of IR? and it is defined as: (h) Finish the definition: Subset H in vector space V is a subspace if:   1 2 (i) Find the inverse of 5 12 2. Find the inverse of the matrix A below:  1 A= 4 −2  1 −1 2 −1  −1 1 3. Suppose A, B and X are n × n matrices, with A, X, and A − AX invertible, and suppose (A − AX)−1 = X −1 B First, explain why B is invertible, then solve the equation for X. If you need to invert a matrix, explain why it is invertible. 4. Show that, if  a 5. Let A =  d g AB is invertible, then so is A (assume A, B are n × n).      b c a + 2g b + 2h c + 2i g h i e f , B =  d + 3g e + 3h f + 3i , and C =  2d 2e 2f . h i g h i a b c If det(A) = 5, find det(B), det(C), det(BC). 6. Assume that A and B are row equivalent, where:   1 2 −2 0 7  −2 −3 1 −1 −5  , A=  −3 −4 0 −2 −3  3 6 −6 5 1  1  0 B=  0 0 0 1 0 0  4 0 −3 −3 0 5   0 1 −4  0 0 0 (a) State which vector space contains each of the four subspaces, and state the dimension of each of the four subspaces: (b) Find a basis for Col(A): (c) Find a basis for Row(A): (d) Find a basis for Null(A): 1 7. Determine if the following sets are subspaces of V . Justify your answers.     a  • H =  b  , a ≥ 0, b ≥ 0, c ≥ 0 , V = IR3   c    a + 3b        a − b   • H=  , a, b in IR , V = IR4  2a + b       4a • H = {f : f 0 (x) = f (x)} , V = C 1 (−∞, ∞) (C 1 is the space of differentiable functions where the derivative is continuous). • H is the set of vectors in IR3 whose first entry is the sum of the second and third entries, V = IR3 . 8. Prove that, if T : V 7→ W is a linear transformation between vector spaces V and W , then the range of T , which we denote as T (V ), is a subspace of W . 9. Let H, K be subspaces of vector space V . Define H +K as the set below, and see if H +K is a subspace (check all parts of the definition). H + K = {w | w = u + v, for some u ∈ H, v ∈ K} 10. Let A be an n × n matrix. Write statements from the Invertible Matrix Theorem that are each equivalent to the statement “A is invertible”. Use the following concepts, one in each statement: (a) Null(A) (b) Basis (c) Rank (d) det(A) 11. Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.  12. Show that 1, 2t, −2 + 4t2 is a basis for P2 . 13. Let T : V → W be a 1 − 1 and linear transformation on vector space V to vector space W . Show that if {T (v1 ), T (v2 ), T (v3 )} are linearly dependent vectors in W , then {v1 , v2 , v3 } are linearly dependent vectors in V . 14. Use Cramer’s Rule to solve the system: 2x1 −3x1 +x2 x2  15. Let A = −6 −3 12 6 +x3 +2x3 =7 = −8 = −3  , and w = [2, 1]T . Is w in the column space of A? Is it in the null space of A? 16. Prove that the column space is a vector space using a very short proof, then prove it directly by showing the three conditions hold. 17. If A, B are 4 × 4 matrices with det(A) = 2 and det(B) = −3, what is the determinant of the following (if you can compute it): (a) det(AB), (b) det(A−1 ), (c) det(5B) (d) det(3A − 2B), (e) det(B T ) 18. True or False, and give a short reason: (a) If det(A) = 2 and det(B) = 3, then det(A + B) = 5. (b) Let A be n × n. Then det(AT A) ≥ 0. 2 (c) If A3 is the zero matrix, then det(A) = 0. (d) IR2 is a two dimensional subspace of IR3 . (e) Row operations preserve the linear dependence relations among the rows of A. (f) The sum of the dimensions of the row space and the null space of A equals the number of rows of A. 19. Let the matrix A and its RREF,  1  3 A=  −3 2 RA , be given as below:   1 1 7 2 2  0 0 9 3 4   RA =   0 1 −5 −2 3  0 2 14 4 2  0 0   1  0 0 1 0 0 3 4 0 0 1 1 0 0 0 1 0 0 0 0 1 0  −4 7   −5  0 so that the columns of A are a1 , · · · , a5 . Similarly, define Z and its RREF, RZ , as:  4 5 3 4  5 6 5 −3 Z=  10 −3 9 −106 4 10 2 44      1  0 Rz =   0 0 Label the columns of Z as z1 , · · · , z4 . (a) Find the rank of A and a basis for the column space of A (use the notation a1 , etc.). Similarly, do the same for Z: (b) You’ll notice that the rank of A and Z:  1 1  3 0   −3 1 2 2 A is the rank of Z. Here   2 4 5 3 1  0 4 5 6 5  →  0 3 10 −3 9  2 4 10 2 0 is a row reduction using some columns of  0 0 −1 2 −1 1 0 1 3 0   0 1 2 0 2  0 0 0 0 0 Are the subspaces spanned by the columns of A and Z equal? (c) Let B and be the set of basis vectors used for the column spaces of A found in (a). Find the change of coordinates matrix PB that changes the coordinates from B to the standard basis, then find the coordinates of z1 with respect to B (Hint: The second part does not rely on the first). (d) Find the coordinates of z4 using the basis vectors in z1 , z2 , z3 . 20. Short Answer: (a) Define the kernel of a transformation T : (b) Define the dimension of a vector space: (c) We said that Pn is isomorphic to IRn+1 . What is the isomorphism? (d) If C is 4 × 5, what is the largest possible rank of C? What is the smallest possible dimension of the null space of C? (e) If A is a 4 × 7 matrix with rank 3, find the dimensions of the four fundamental subspaces of A. (f) Show that the coordinate mapping (from n−dimensional vector space V to IRn ) is onto. 21. Let A be m × n and let B be n × p. Show that the rank(AB) ≤ rank(A). (Hint: Explain why every vector in the column space of AB is in the column space of A). 22. Let T : IRn → IRm be a linear transformation. 3 (a) If T is one-to-one, what is the dimension of the range of T ? (b) What is the dimension of the kernel of T if T maps IRn onto IRm ? Explain. 23. Find the determinant of the matrix A below:    A=   4 0 6 0 0 8 1 8 8 8 8 0 8 8 2 8 0 8 3 0 5 0 7 0 0       24. Let A, B be given below. Form the matrix product AB, if defined:     −1 1 2 0 −1 A= B= 2 1  1 1 0 1 2 25. Given the matrix A, B below:  1 A =  −1 3 2 1 2  3 0  0  1 B= 2 1  −1 2 1 1  0 1 (a) Compute only the (2, 3) entry of AB: (b) Compute only the (3, 2) entry of AB T : (c) Compute B − 3I3 : (d) Compute C23 for matrix A (that’s the (2,3) cofactor). 26. If A is the 2 × 3 matrix below, find a matrix C so that AC = I, but note that C is not the inverse of A. To simplify your computations, I’ve given you one form for C that you might use.     c11 c12 −1 2 −1 A= C =  c21 c22  6 −9 3 0 0 27. Suppose A is n × n with the property that Ax = ~0 has only the trivial solution. Without using the invertible matrix theorem, explain directly why the equation Ax = b must have a solution for every b. 28. Explain why the columns of A2 span IRn whenever the columns of A are linearly independent. (Hint: You might think about whether or not A2 must be invertible). 29. Suppose subspace H is the span of the two vectors below in set B:     2   1 B =  0  ,  3  = {v1 , v2 }   0 0 (a) Does B span IR3 ? Why or why not? (b) Find [v1 ]B (c) If c = (3, 3, 0), find [c]B 4