Let a subset W be the set of all vectors in R² such that |x₁|=|x₂|. Apply the theorem for conditions for a subspace to determine whether or not W is a subspace of R². According to the theorem of conditions for a subspace, the nonempty subset W of the vector space V is a subspace of Vit and only if it satisfies the following two conditions: (i) If u and v are vectors in W, then u + v is also in W. (ii) If u is in W and c is a scalar, then the vector cu is also in W. Select the correct choice below. OA. W is a subspace of R2 because it satisfies both of the conditions. O B. W is not a subspace of R2 because both conditions (i) and (ii) fail. OC. W is not a subspace of R2 because condition (ii) while condition (i) satisfied. ⒸD. W is not a subspace of R² because condition (i) fails while condition (ii) is satisfied.

. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v EV can be represented uniquely as v = v₁ + v2 where V₁ € W₁, V2 € W2. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V = W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that B₁ U B2 is a basis of V.

Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1. Show that 1 (v, w1) + 2 (v, w2) + 3 (v, w3) + ...+ k (v, wk) cannot be a negative number.

Let be the set A = {(2, –2,6), (1,2,0), (3,0,6)}, of vectors of R a) Determine the vector space w (generic element) that is spanned by the set A. b) Obtain the standard base and the dimension of w. c) State whether the vector (-1,1, –3) belongs to the space W.