- #1

Mark53

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## Homework Statement

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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