State whether the following statements are true or false, with reasoning:
a. Let V := R 2 . Let + denote the usual (vector) addition on R 2 . Define a scalar multiplication ∗ by assigning α ∗ (x, y) = (αy, αx), for any α ∈ R and (x, y) ∈ R 2 . Then (V, ∗, +) is not a vector space.
b. Now let V = R 2×2 , the set of all 2 × 2 real matrices. Let + denote the usual matrix addition (as mentioned in class). But define a scalar multiplication
c. by assigning α • A = O, for any α ∈ R, and O is the zero matrix in R 2×2 . Then, (V, •, +) is a vector space.
