Linear Transformation of Dependent set. Let $V$ and $W$ be vector spaces and let $T: V\to W$ be a linear transformation. Let $\{v_1, v_2,\ldots, v_p\}$ be a linearly dependent set of vectors in $V$. Show that $\{Tv_1, Tv_2,\ldots, Tv_p\}$ is also linearly dependent.

This is a direct consequence of linearity. I mean, it's almost by definition. Independence can be established by looking at a single equation- $c_1v_1 + \cdots + c_nv_n = 0.$ Applying any linear transformation to that equation wont affect its solutions.
