5. For two 2 × 3 real matrices A = [ai,j], B = [bi,j] € Mat(2 × 3, R), define d(A, B) :: := max{|ai,j - bij|: i = 1, 2, j = 1, 2, 3} (a). Show that d is a metric on Mat (2 x 3, R). (b). Let X = {A E Mat(2 × 3, R): rows of A are linearly indepedent Show that X is an open subset of Mat(2 × 3, R) under the metric d.

Let U be a subspace of R", and let S = { ū, v, w } be a set of three non-zero vectors in R”. Fill in the blanks so that the following statements are true. If you believe you don't have enough information to conclude anything for any of the cases, choose the answer "???". Note that there is no intended relationship between the parts. In each case, assume only what the statement says to assume. If SCU, then dim(U) > (Enter the largest integer that makes the statement necessarily true.) If S is linearly independent and S C U, then dim(U) 3. If S' is a spanning set for U, then dim(U) 3. If span(S) = U then S If S is a basis for U, then S linearly independent. linearly independent.