Base change/Standard basis in R^4/Example/Exercise

Consider the standard basis ${}e_{1},e_{2},e_{3},e_{4}$ in ${}\mathbb {R} ^{4}$ and the three vectors

${\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.$

Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector, as shown in the base exchange theorem. Can one take any standard vector?

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