- #1
Mark53
- 93
- 0
Homework Statement
[/B]
1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}.
(a) Prove that {Av1, . . . , Avk} is linearly independent.
(b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what conditions on m and n will {Av1, . . . , Avk} be a basis for Rm?
The Attempt at a Solution
a)
we know that c1V1+...+CnVn=0 as it is linearly independent
suppose that
C1AV1+...+CnAVn=0 and that A is an invertible matrix since the null space is 0 which means we can multiply both sides by A^-1 which gives c1V1+...+CnVn=0 which means that there is a trivial solution and that it is linearly independent
Is this correct?
b)
Im unsure on how to get started on this question
Thanks for any help