Question

If v1 and v2 are linearly independent vectors is av1 and av2 also independent

Answer 1

Assume {Av1,...,Avk} is not linearly independent, then Av1=a2(Av2)+...+ak(Avk), where ai are scalars from the field and not all ai are equal 0. but then

Av1=A(a2v2+...+akvk)

muliplying by A−1 we get v1=a2v2+...+akvk and {vi} for a set that is not linearly independent.