W = {[a b c] : c = a + b} 5. Use Theorem 4.3 to determine which of the following

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W = {[a b c] : c = a + b} 5. Use Theorem 4.3 to determine which of the following are subspaces of M_2,2. (a) The set of all upper triangular 2 times 2 matrices. (b) The set of all 2 times 2 matrices such that the sum of the entries equal 0. (c) The set of all 2 times 2 matrices with integer entries. (d) The set of all 2 times 2 matrices A such that det A = 0. 6. Use Theorem 4.3 to determine which of the following are subspaces of P, the vector space of all polynomials. (a) The set of all polynomials of degree 3. (b) The set of all polynomials of the form p(x) = ax^3 where a is any real number. (c) The set of all polynomials p(x) = a_nx^n +...+ a_1x + a_0 such that a_3 = a_0. (d) The set of all polynomials, p(x), with derivatives at x = 3 equal to 0 (i.e. p'(3) = 0). 7. Use Theorem 4.3 to determine which of the following are subspaces of D, the vector space of all differentiable functions on (-infinity, infinity). (a) The set of all differentiable functions, f, such that f(0) = f(1). (b) The set of all differentiable functions, f, such that f'(x) = 3f(x). (c) The set of all differentiable functions, f, such that f'(3) = 0. (d) The set of all differentiable functions, f, such that f'(0) = 1. 8. Use the subspace test to prove that each of the following are subspaces. (a) Let w_1 = {[x y z] R^3 : x + 2y + 3z = 0}. Prove W is a subspace of R3. (b) Let D_2 = {A M_2,2 : A is a diagonal matrix}. For the definition of diagonal matrix, see exercise 16 of problem set 1.5. Prove D_2 is a subspace of M_2,2. (c) Let Z_3 = {p(x) P: p(3) = 0}. Prove Z_3 is a subspace of P. (d) Let A be a fixed n times n matrix and let E_4 = {x R^n : Ax = 4x}. Prove E_4 is a subspace of R^n. 9. Suppose V is a vector space and w_1, and W_2 are subspaces of V. Determine whether the following three subsets of V must be subspaces of V. Use Theorem 4.3 to prove the subset is a subspace or give a counterexample to demonstrate the subset need not be a subspace. (a) W_1 W_2 (b) W_1 W_2 (c) W_1 + W_2 = {w_1 + w_2 : w_1 w_1, w_2 W_2}

Explanation / Answer

What is the Theorem 4.3

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