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Math 333 First Exam (9 Oct 2014) Name: If you use a theorem indicate what it is. instructed otherwise). Good Luck! Show all work (unless 0. Circle True or False without explanation: ( T or F ) Every non-zero vector v in V belongs to some basis of V . ( T or F ) The spaces P3 and M2×2 are isomorphic (to each other). ( T or F ) If {v1 , . . . , vn } is independent then so is {T v1 , . . . , T vn } (for a linear T ). ( T or F ) If A is a square matrix and A2 = 0 then A = 0. ( T or F ) For any b in Z8 , the equation 6x = b has a solution x in Z8 . 1. Find a basis and the dimension for the space A3×3 of all 3 × 3 antisymmetric matrices. Start by filling-in the six blanks in the general from of such a matrix. A3×3 := a b c : a, b, c ∈ R . Do not prove anything. 2. Quickly and decisively show that the set In×n of all invertible n × n matrices is not a linear subspace of Mn×n (for any natural n). 3. Carefully prove that the following subset W of F is a subspace: W := {f ∈ F : f (2x) = f (x) for all x ∈ R}, (Plainly, W consists of all functions on R unchanged by the substitution x 7→ 2x.) 4. Give a bullet-proof verification of independence of {1, ex , e−x } in F . (If too hard, do {ex , e−x } for partial credit.) 5. Consider two transformations T and S on Mn×n given by 1 S(A) := (A + AT ) 2 1 and T (A) := (A − AT ). 2 a) Carefully verify that S is linear. b) Compute and simplify the formula for the composition S ◦ T S ◦ T (A) = . . . 6. Consider T : P2 → P2 given by T (p(x)) := p(x) + p′ (x). a) Compute the matrix [T ]B←B when B = {1, x + 1, x2 }. b) Determine the kernel of T . c) Based on b), can you tell readily if T is onto. Explain. 7. Prove that if T : V → W is linear and ker(T ) = {0} then T is one-to-one. 8. A binary code encodes x = (x1 , x2 , x3 )T to (b1 , b2 , b3 , b4 , b5 )T := Gx where G is 5 × 3 matrix of rank three and with rows satisfying r1 + r2 + r3 = r4 + r5 and r2 = r4 + r3 . Find the conditions for (b1 , b2 , b3 , b4 , b5 )T to be an uncorrupted message (i.e., b = Gx for some x). Explain your reasoning!