Answer Happy

5. (16 pts) For each of the statements below, determine whether it is True or False. Each correct answer will be awarded 2 points. Each incorrect assertion of True / False will receive a deduction of 1 point. There is no penalty for leaving an answer blank. The minimal overall score for this problem is 0 points. For this problem, you are not required to justify your answers. (a) Let A be a 15 x 15 matrix. If A is invertible, then the span of columns of A must contain a subspace of dimension 4. (b) The dimension of the null space of a 19 x 27 matrix must be less than or equal to 8. (c) If a matrix contains 45 columns which are linearly independent, then it must have at least 45 rows, (d) Suppose A is a 100 x 100 matrix of rank 99. Then A has rank 98. (e) Row equivalent matrices must have the same column space. (f) Let A and B be 5 x 12 matrices, then the rank of the block matrix (AB) is necessarily equal to r(A) +r(B). (Here, r(A), r(B) denote the ranks of A, B, respectively.) (?) Suppose Vi, v2 are two vectors in R", such that || v1 + v2|| = || 1 11? + ||v2|1%. Then vi is orthogonal to V2 (h) If x is an eigenvector of two m x n matrices A and B, then it is an eigenvector of A+ B.