How to prove linear independence of vectors given linear independence of their mapping?

Question

I have the question:

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Let $h: V\to W $ be any linear map.

Prove that: for any $w_1,...w_k$ in $V$,

if: $h(w_1),...h(w_k)$ in $W$ are linearly independent,

then: $w_1,...w_k$ are also linearly independent.

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I have been trying to figure this out, and have found some answers going from $w_1,...w_k$ to $h(w_1),...h(w_k)$, but I am finding that I really struggle with proofs. Any help is greatly appreciated!

Answer 1

Let $$c_1 W_1+....+c_2 W_2+c_k W_k =0$$

Then $$T (c_1 W_1+....+c_2 W_2+c_k W_k) = T(0)=0$$

But $$T (c_1 W_1+c_2 W_2+....+c_k W_k)= c_1 T(W_1)+c_2 T(W_2)....+c_kT(W_k)$$

Since $ T(W_1),T(W_2),... ,T(W_k)$ are linearly independent, all $c_i$ s are $0$

Thus $W_1, W_2, ...,W_k$ are independent.