I am new to proofs and here is my attempt at the following problem. Is it correct? Is it wrong? Any feedback?
Suppose T is a linear transformation and that $T(\vec{u_1})$ and $T(\vec{u_2})$ are linearly independent. Prove that $\vec{u_1}$ and $\vec{u_2}$ must be linearly independent.
My attempt: Proof by contradiction.
Suppose $\vec{u_1},\vec{u_2}$ are linearly dependent. Then $\vec{u_1}=k\vec{u_2}$ where $k \in \mathbb{R}$.
Because of linear independence, $aT(\vec{u_1)} = bT(\vec{u_2}) \implies a=b=0$
Then $aT(k\vec{u_2})=bT(\vec{u_2}) \implies akT(\vec{u_2})=bT(\vec{u_2}) \implies k = \frac{b}{a}$
However, this is impossible because $a=0$ therefore there is no possible value for $k$ and this means that $\vec{u_1}$ and $\vec{u_2}$ must be linearly independent.