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1. Determine Whether the set with given operation defines a vector space. If not, indicate which law(s) fail. (a) V = with standard matrix addition and scalar multiplication (b) V consists of all continuous functions f(x) on the interval [0, 1) such that f(0) = 0 with standard function addition and scalar multiplication. (V = {J(x) € C(0,11/5(0) = 0}). (c) V = {(x, fx)(r € IR} with the standard vector addition and scalar multiplication. (d) Let V be the set C3 with the standard vector addition, but with scalar multiplication defined by 2 az a y = oz z ag (e) V is the set of all fifth-degree polynomials with standard operations. 2. Determine whether the given subset W is a subspace of the vector space V. Justify your answer. (a) V = R3 and W = {(a,0, a - b)|a, b € R} (b) V = C10, 1] and W = {J(x) € C10, < 0). (c) V = CS and W = 6 5 3. Find the redundant vectors, if any in the following sets. (a) S = {r,5x}} (b) S = (2,2-2,27,1+224 (c) S = 4. Assume V is a vector space of dimension 72 and S = (U1, V vit CV. Answer True or False for the following statements. (a) S is either a basis or contains redundant vectors. (b) A linearly independent set contains no redundant vectors. (c) If V = (zy. vg) and dim V = 2, then the set is a linearly dependent set. (d) A set of vectors containing the zero vector is linearly independent set. (e) Every vector space is finite-dimensional. (f) The set of vectors { : i : 0 1 in C2 contains redundant vectors.

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