Question

Determine whether the following vectors form a basis of ${\mathbf{ℝ}}^{\mathbf{4}}$; $\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ 2\\ 4\\ 8\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ -2\\ 4\\ -8\end{array}\right]$.

Short answer

The given vectors $\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 2\\ 4\\ 8\end{array}\right],\left[\begin{array}{c}1\\ -2\\ 4\\ -8\end{array}\right]$form a basis of${\mathrm{ℝ}}^4$.

Step-by-step solution

Definition of the basis of ℝn

The vectors $\overrightarrow{v_1},\overrightarrow{v_2},.....,\overrightarrow{v_n}$in ${\mathrm{ℝ}}^n$form a basis of${\mathrm{ℝ}}^n$if (and only if) the matrix

$A=\left[\overrightarrow{v_1}\overrightarrow{v_2}...\overrightarrow{v_n}\right]$ is invertible.

Finding the determinant

Here we have $A=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 2& -2\\ 1& 1& 4& 4\\ 1& -1& 8& -8\end{array}\right]$

$$\begin{gathered} \Rightarrow \left|A\right|=1\left[\begin{array}{ccc}-1& 2& -2\\ 1& 4& 8\\ -1& 8& -8\end{array}\right]-1\left[\begin{array}{ccc}1& 2& -2\\ 1& 4& 4\\ 1& 8& 8\end{array}\right]+1\left[\begin{array}{ccc}1& -1& -2\\ 1& 1& 4\\ 1& -1& -8\end{array}\right]-1\left[\begin{array}{ccc}1& -1& 2\\ 1& 1& 4\\ 1& -1& 8\end{array}\right] \\ \Rightarrow \left|A\right|=72-0-12-4=56 \\ \Rightarrow \left|A\right|=56\left(\ne 0\right) \end{gathered}$$

Since the matrix A is invertible, then the vectors $\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 2\\ 4\\ 8\end{array}\right],\left[\begin{array}{c}1\\ -2\\ 4\\ -8\end{array}\right]$ form a basis of ${\mathrm{ℝ}}^4$.

Final Answer

The given vectors$\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 2\\ 4\\ 8\end{array}\right],\left[\begin{array}{c}1\\ -2\\ 4\\ -8\end{array}\right]$ form a basis of${\mathrm{ℝ}}^4$.