How to show a linearly independent subset of one vector space is linearly independent after a linear transformation to another vector space

Question

If there exists a linear transformation from vector space $U$ to $V$, $T:U \rightarrow V$. Show that $T(X)$ is a L.I subset where $X \subseteq U$ and $X$ is L.I.

Not sure how to show this, anyone have any ideas?

Answer 1

This is just not true. consider $T:\mathbb R^2 \to \mathbb R^2$, given by $$\begin{pmatrix}1&-1\\0&0 \end{pmatrix}$$

and take $X=\{(1,0),(0,1)\}$ for a counterexample.

In the case where $T$ is injective on the other hand, it has trivial kernel.

Consider any linear combination $$a_1(Tx_1)+\dots a_n (Tx_n)=0$$ and use linearity to show that this linear combination has to have $a_1 =\dots =a_n=0$ for any linearly independent $x_1, \dots x_n$.