# Describing a span of vectors in $\mathbb{R^4}$.

I am given that $\mathbf{v}_1 = \begin{pmatrix}1\\ 3 \\ 1 \\ 1\\ \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix}1\\ -1 \\ 6 \\ 2\\ \end{pmatrix}$ in $\mathbb{R^4}$. How would I describe $\mathrm{Span}\{\mathbf{v}_1,\mathbf{v}_2\}$? I know that $[\mathbf{v}_1\; \mathbf{v}_2]$ row reduces to $\begin{pmatrix}1 & 0\\ 0 & 1\\ 0 & 0\\ 0 & 0 \end{pmatrix}$, but is there a "nontrivial" way to actually describe $\mathrm{Span}\{\mathbf{v}_1,\mathbf{v}_2\}$ beyond saying that it is all of the linear combinations of $\mathbf{v}_1$ and $\mathbf{v}_2$? For example, is there a geometric description of $\mathrm{Span}\{\mathbf{v}_1,\mathbf{v}_2\}$? I know you can usually describe it as a plane in $\mathbb{R^3}$, but I am unsure of how to give a meaningful description in $\mathbb{R^4}$.

Edit: (is this right?) $\mathrm{Span}\{\mathbf{v}_1,\mathbf{v}_2\}$ is a plane in $\mathbb{R^4}$ through the origin, because neither vector in this problem is a multiple of the other.

• $\mathbb{R^2}$ is to $\mathbb{R^4}$ as $\mathbb{R^1}$ is to $\mathbb{R^3}$. – JonMark Perry Feb 5 '15 at 5:48

• So $\mathrm{Span}\{\mathbf{v}_1,\mathbf{v}_2\}$ is just a plane in $\mathbb{R^4}$ through the origin, because neither vector in this problem is a multiple of the other. Is that right? – fancynancy Feb 5 '15 at 5:59