Let \[\vec{v}_1=\begin{pmatrix} 1 \\ 2 \\ -1 \\ 0 \end{pmatrix}, \quad \vec{v}_2=\begin{pmatrix} 4 \\ 8 \\ -4 \\ -3 \end{pmatrix}, \quad \vec{v}_3=\begin{pmatrix} 0 \\ 1 \\ 3 \\ 4 \end{pmatrix}, \quad \vec{v}_4=\begin{pmatrix} 2 \\ 5 \\ 1 \\ 4 \end{pmatrix}.\] Find a basis of the vector subspace \(\langle\{\vec{v}_1,\vec{v}_2,\vec{v}_3,\vec{v}_4\}\rangle \subset \mathbb{R}^4.\)

Show that a square matrix \(\mathbf{A}\) is invertible if and only if its column vectors are linearly independent.

We consider \(\mathsf{F}([0,1],\mathbb{R}).\) Show that the functions \(x,\sin x,\cos x\) are linearly independent.

Let \(V\) be a vector space over \(\mathbb{C}\) with \(\dim_{\mathbb{C}}(V)=n.\) Show that \(V\) can also be interepreted as a vector space over \(\mathbb{R}\) with \(\dim_{\mathbb{R}}(V)=2n.\)