300, MATH Second Exam REVIEW QUESTIONS
You should also look at the supplementary exercises for Chapter 4, problems 1-11
1. Let u = and u + v.
2 1
and v =
-2 1
, and A =
4 5
3 3
. Let S be the parallelegram with vertices at 0, u, v,
Compute the area of S. Compute a 2. Let A = d g the area of the image of S under the mapping x Ax. b c a + 2g b + 2h c + 2i g e f , B = d + 3g e + 3h f + 3i , and C = 2d h i g h i a h i 2e 2f . b c
If det(A) = 5, find det(B), det(C), det(BC). 3. 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 (a) Find a basis for Col(A) (b) Find a basis for Row(A) (c) Find a basis for Null(A) 4. Let A be an m n matrix of rank k. What are the dimensions of the 4 fundamental subspaces? 5. Determine if the following sets are subspaces of V . Justify your answers. a V =I 3 R H = b , a 0, b 0, c 0 , c a + 3b a-b , a, b in IR , V = IR4 H= 2a + b 4a H = {f : f (x) = f (x)} , V = C 1 [I R] H is the set of vectors in I 3 whose first entry is the sum of the second and third entries, V = IR3 . R For the following question, assume A is n n, and let be a given eigenvalue of A. Show directly (without the use of theorems) that the following is a subspace of IRn : H = {x : Ax = x} 6. Consider a homogeneous system of seven linear equations in nine variables. Suppose that there are two solutions of the system that are not multiples of one another, and all other solutions are linear combinations of these solutions. Will the associated nonhomogeneous system have a solution for every possible choice of constants on the right sides of the equations? Justify your answer carefully. 7. Let A be m n. Justify: dim(Col(A)) + dim(Null(AT )) = m 8. Use the previous answer to justify the following statement: Ax = b has a solution for every b IRm if and only if the equation AT y = 0 has only the trivial solution. 1
1 0 4 0 1 -3 B= 0 0 0 0 0 0
0 3 0 5 1 -4 0 0
9. Prove that, if T : V W is a linear transformation between vector spaces V and W , then the range of T is a subspace of W . 10. Let A be an n n matrix. Write statements from the Invertible Matrix Theorem that are equivalent each to the statement "A is invertible". Use the following concepts, one in each statement: (a) Null(A) (b) Basis (c) Rank 11. Let B = {u1 , u2 }, where u1 = 1 2 , u2 = 3 -1
If [x]E = (3, 0), what is [x]B ? If [x]B = (4, -2), what is [x]E ? 12. Use Cramer's Rule to determine for which values of s the following system has a solution, then write the solution in terms of s: 5x1 + 3sx2 2sx1 + 2x2 =1 =3
13. Find a basis for the null space and column space of A using Z2 arithmetic. Determine the rank of the matrix (again, with respect to Z2 ). Answer the same questions, but use the standard real number arithmetic. 1 0 0 1 1 A= 0 1 1 1 1 1 1 0 1 0 1 2 4 3 14. Let v 1 = 0 , v 2 = 1 , v 3 = 2 , w = 1 . Is w in {v 1 , v 2 , v 3 }? Can w be written as a -1 3 6 2 linear combination of v 1 , v 2 , v 3 ? 15. 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 ) 16. We said that, if V is a vector space with n basis vectors, then V is isomorphic to IRn . What is the isomorphism? In general, what is an isomorphism? 17. 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. 18. Let IRn be represented by three sets of basis vectors, B = {bi }i=1 , C = {ci }i=1 , and the standard basis n {ei }i=1 . Draw a diagram that relates these three versions of IRn , and write down the mappings that relate each space to the other two. 19. Let B = 3 4 -5 -6 and from [x]B to [x]C . , and C = 4 5 6 7 . Write down the matrices that take [x]C to [x]B
n n
20. Let H be a subspace of V , and let T be a linear mapping of vector space V to vector space W , where V, W are finite dimensional. Show that T (H) is a subspace of W , and show that dim(T (H)) dim(H). What do we need to say about T in order to get equality? 21. (a) Show that 1, 2t, -2 + 4t2 is a basis for P3 . (b) Find the coordinates of 3 - t + 2t2 in terms of this basis. 2 a 22. Show that the rank of uv T 1 if u = -3 and v = b . (Hint: Multiply out the matrix). 5 c
2