true or false

(a) Given A E Mat2x3(R), the function LA is a function from Pi (R) to P2(R). T F (b) The zero element in a field always has a multiplicative inverse. T F (c) Given two subspaces W1, W2 C V, if dim(W1) + dim(W2) = dim(V), then WinW2 = {0}. TF (d) Let V be finite-dimensional and let G, L C V. If span(G) = V and L is linearly indepen dent, then L has at most as many elements as G. T F (e) Suppose that dim(V) = n. Let be T: V - V a linear transformation of rank k, let B be a basis for V, and let A be similar to s. Then the rank of LA : F - F" is also k. T F (f) If V is an inner product space over a field F = R or C, x E V is any vector, and a E F is any scalar, then |(x, ax)| = Hall . Ilaxll. T F (g) The determinant function det: Mat3x3(Z/23Z) - Z/23Z is onto. T F (h) If dim(V) = n, a linear transformation T: V - V is given, and fr(t) factors into linear terms over F, then T is diagonalizable. T F (i) If A1, 12 are different eigenvalues for T: V - V then Ex, + Ex, is a direct sum. T F (j) Given any vector space W, there exists a vector space V and a linear transformation T: V - W such that Im(T) = W. TF