Linear Algebra question. Please help?

Question

Let A be an mxn matrix and b1,...,bk is and element of R^m. Suppose {b1,...,bk} is linearly independent. Suppose that v1,...,vk is an element of R^n can be chosen so that Av1=b1,...,Avk=bk. Prove that {v1,...,vk} must be linearly independent.

Answer 1

initial assuption is that {b1,...,bk} is linearly independent.

so we can write cc1*b1 + cc2*b2 + ... +cck*bk = 0 equation has only the trivial solution cc1 = 0, cc2 =0, ... cck=0; cc1, cc2,...cck are scalars.

substituting b1 = Av1 ...

now for scalars c1, c2, c3 ... ck
let us assume
A(c1*v1 + c2*v2 + ... ck*vk) = 0
or, A(c1*v1) + A(c2*v2) + ... A(ck*vk) = 0
or, c1*(Av1) + c2(Av2) + ... +ck(Av3) = 0
or, c1*b1 + c2*b2 + ...+ck*bk = 0

so c1 = 0, c2 = 0, c3 = 0
(c1*v1 + c2*v2 + ... ck*vk) = 0+0+...+0 = 0

so v1, v2, ...vk are Linearly independent

Answer 2

Easy exercise for HOMEWORK.
Hint: suppose some linear combnation of the v_i are zero. Now apply A.

Answer 3

The only way to learn is to work it out for yourself... Or in other words, do your own homework...