1. Justify your answers to the following with brief explanations. (a) True or fa

Question

1. Justify your answers to the following with brief explanations.
(a) True or false?
i. If there columns of A are linearly independent, then Ax = b has exactly one solution for
every b.
ii. A 9 x 10 matrix never has linearly independent columns.
iii. The column space of a 2 x 2 matrix is the same as its row space.
iv. The column space of a 2 x 2 matrix has the same dimension as its row space.
(b) Let A be a 64x17 matrix with rank 13: How many independent vectors satisfy Ax = 0? How
many independent vectors satisfy AT y = 0?

2. Find a matrix A that has W as its row space, and (extra credit) a matrix B that has W as its
nullspace, if W is the subspace spanned by

Explanation / Answer

1(a)(i) True . Since rank(A)=rank([A|b])=n, n is the number of columns in this case.

(ii)True. By rank nullity theorem.

(iii) False. Only true when matrix is a symmetric matrix.

(iv) False. We can have matrix with row vectors dependent but column vectors independent.

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