Math 136 Spring 2017: Assignment 5 (Due Wednesday, June 7, 8pm) Topics: Section 3.1, 3.2 operations on matrices, linear mappings. 1. Let A = \u0014 \u0015 \u0014 \u0015 \u0014 \u0015 3 1 4 2 2 1 4 1 1 ,B= ,C= . 1 1 0 1 3 1/2 0 2 2 (a) Compute 2A B. (b) Compute A(B T + C T ). (c) Compute BAT + CAT . 2. Determine if there exists scalars t1 , t2 , t3 such that \u0014 \u0015 \u0014 \u0015 \u0014 \u0015 \u0014 \u0015 2 1 4 1 3 3 0 2 t1 + t2 + t3 = 1 1 2 1 3 2 1 1 3. Linear mappings. (a) Let f (x1 , x2 , x3 ) = (x1 + x2 , x3 , 0). i. Is f a linear map? Justify ii. If f is linear find its standard matrix A = [f ]. iii. The solution set S of the system [A |~0 ] is known to be a subspace of R3 . (Notice that S = {~x : f (~x) = ~0}.) Find a basis for the set S. (b) Let T : R2 R2 be defined as T (~x) = proj~a x ~ . (~a 6= ~0) i. Is T a linear map? Justify ii. If T is linear where ~a = (2, 1) find its standard matrix B = [T ]. iii. The solution set S of the matrix equation B~ x = ~0 is known to be a subspace of R2 . Find a basis for the set S. (c) Consider the function L : R2 R3 defined as L(x1 , x2 ) = (0, x1 + 2, x2). Is L a linear map? Justify. 4. Let the line L be defined as L = {(x1 , x2 ) : x1 5x2 = 0}. The reflection reflL : R2 R2 into the line L is known to be linear. Determine its standard matrix, [reflL ]. 1 2 5. Let A = 1 2 . 1 2 (a) Find a basis for the solution set of the system [ A | ~0 ]. (b) Find a basis for the solution set of the system [ AT | ~0 ]. 6. Let L and M be linear mappings from Rn to Rm , and k R. (a) Prove that L + M and kL are both linear mappings. (b) Prove that [kL + M ] = k[L] + [M ]. (Recall that, by definition kL(~ x) + M (~ x) for all x ~ .) 1 of the function kL + M , (kL + M )(~ x) =