Consider the 2 X 2 matrices A1 = 1 1 A2 = 0 1 A3 = 0 0 A4 = 1 0 1 1 1 0 () Argue whether these four matrices are linearly independent or not in the vector space of 2 X 2 real matrices. (ii) Find the dimension of span(A1, A2, A3, A4).

Let A B denotes the tensor product of matrices Amxn and Bpxq defined by the block matrix A B = a12B 922B a11B a21 B : : ami B am2 B : ain B a2n B : amn B mpxnq (a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr. Prove that (AB)(C > D) = ACⒸ BD (b) Let Amxm and B₁xn be nonsingular matrices. - Prove that (AB) is nonsingular. - Find the inverse matrix of (AB) (c) Prove that the following equality applies for any Amxm and Bnxn square matrices trace(AB) = trace(A) * trace(B)

Let x=(4,3, 1, 1) and y (8, 2, 7, 1). Determine which of the following statement(s) is/are true: (i) xy is a scalar (ii) xy" is a matrix (iii) yx is a matrix (iv) xyx is a vector O a. (ii), (i) and (iv) only O b. (i), (ii) and (iv) only Oc. All of the above O d. (i), (ii) and (iii) only.

3= b, ,b2} and C= (c,.c2} be bases for a vector space V, and suppose b, = - 6c, + 5c2 and b2 = - 4c, +2c2. a. Find the change-of-coordinates matrix from B to C. b. Find (x]c for x = - 5b, + 3b2. Use part (a). a. P = C-B b. (xlc (Simplify your answers.)

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Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. [:::) 1 2 0 - 3 3 6 0 - 9 x=x+x+xO X= X2 + X3 +X4 (Type an integer or fraction for each matrix element.)