# 1. The augmented matrix of a linear system has the form 1 2 1 a ...

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1. The augmented matrix of a linear system has the form 1 2 1 a -1 3 1 1 b 3 -5 5 1 C 2 -2 4 2 d (a) Can you determine by inspection whether the determinant of the coefficient matrix is 0? Explain. (b) Can you decide by inspection whether the linear system has a unique solu- tion for every choice of a, b, C. d? Explain. (c) Determine the values of a, b, c. d for which the linear system is consistent and inconsistent. (d) If a - 2, b = 1. C - -1, and d - 4, find the solution set for the linear system. 2. Let 1 1 M1 = 1 M2 = 2 1 M3 = 1 : (a) Show that the set {M1. M2, Ms} is linearly independent. (b) Describe all matrices that can be written as a linear combination of M1, M2, and M3. (c) Find a basis for the space of matrices found in part (b) and give the space's dimension. (d) Give an example of a space that is isomorphic to the space found in (b). Specify the isomorphism between the two spaces. 3. Let c be a fixed scalar and let m(x) 1 p((c)==to (3(1)=(I+C)? = (a) Show that B - {p1(x) p2(x), P3(x)} is a basis for P2. (b) Find the coordinates of f(x) = ao + ax + 02x² relative to B. 4. Let W C R3 (i.e. W is a subset of R³) be the plane with equation I- = 21 i.e. W = {F € R3 on - T2 - T3 - 0} where Fil = T2 T3 (a) Show W is a subspace of the vector space R³ with standard addition and scalar multiplication (b) Find a basis for W and then use the Gram-Schmidt procedure to obtain an orthonormal basis for W. 5. Define a transformation T P2 R by T(p(x)) = fo p(c)de (a) Show T is a linear transformation (b) Compute N(T). Is T one-to-one? (c) Show that T is onto. (d) Let B be the standard basis for P2 and let B' = {1} be a basis for R. Find [T), (e) Use the matrix found in part (d) to compute T1-x2 - - 3x+2). a 6. Let A = for some a. 6 € R. b a (a) Find the eigenvalues and eigenvectors of A. (b) Diagonalize A. That is, specify P and D such that D = P-1AP. 7. Let A be an n x n matrix. Show that det(A7) = det(A) using a proof by induction. (Hint: the base case is not n=1.) 8. Suppose that S = {01,02,23} is linearly independent and W1 = is + in + is 12- w3 U3. Show that T - {ur W2 w3} is linearly independent.

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