Question

Suppose ${\mathbb{R}}^4=Span\left\{v_1,...,v_4\right\}$. Explain why $\left\{v_1,...,v_4\right\}$ is a basis for ${\mathbb{R}}^4$.

Short answer

The set $\left\{v_1,...,v_4\right\}$ is a basis for ${\mathbb{R}}^4$.

Step-by-step solution

State the invertible matrix theorem

Recall the invertible matrix theorem which states that if the square matrix is invertible, then the columns are linearly independent, and the columns form a basis for ${\mathbb{R}}^n$.

State the basis theorem

It is given that$\left\{v_1,...,v_4\right\}$is the set of vectors. It forms a square matrix of the order$4\times 4$as shown below:

$A=\left[\begin{array}{cccc}{\mathbf{v}}_1& {\mathbf{v}}_2& {\mathbf{v}}_3& {\mathbf{v}}_4\end{array}\right]$

As${\mathbb{R}}^4=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\left\{v_1,...,v_4\right\}$, the columns of A span${\mathbb{R}}^4$. Also, the columns are linearly independent and form a basis.

Thus, $\left\{v_1,...,v_4\right\}$ is a basis for ${\mathbb{R}}^4$.