Question

For which value(s) of the constant k do the vectors below form a basis of ${\mathbf{ℝ}}^{\mathbf{4}}$?

$\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right]\mathbf{}\mathbf{,}\mathbf{}\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right]\mathbf{}\mathbf{}\mathbf{,}\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right]\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\left[\begin{array}{c}2\\ 3\\ 4\\ K\end{array}\right]$

Short answer

The given vectors $\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 4\\ \mathrm{K}\end{array}\right]$form a basis of ${\mathrm{ℝ}}^4{\mathrm{ℝ}}^4{\mathrm{ℝ}}^4$for all values of$k\in \mathrm{ℝ}-\left\{29\right\}$ .

Step-by-step solution

Definition of the basis

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

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

Finding the determinant

Here we have

$A=\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 3\\ 0& 0& 1& 4\\ 2& 3& 4& k\end{array}\right]$

$$\begin{gathered} \Rightarrow \left|A\right|=1\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 4\\ 3& 4& k\end{array}\right]-0\left[\begin{array}{ccc}0& 0& 3\\ 0& 1& 4\\ 2& 4& k\end{array}\right]+0\left[\begin{array}{ccc}0& 1& 3\\ 0& 0& 4\\ 2& 3& k\end{array}\right]-2\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 2& 3& 4\end{array}\right] \\ \Rightarrow \left|A\right|=\left(k-25\right)+0+0+\left(-4\right) \\ \Rightarrow \left|A\right|=k-29 \end{gathered}$$

Finding the values of k

Since it is given that the vectors $\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 4\\ \mathrm{K}\end{array}\right]$form a basis of ${\mathrm{ℝ}}^4$then the matrix $A=\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\begin{array}{c}2\\ 3\\ 4\\ k\end{array}\right]$must be invertible.

$$\begin{gathered} \Rightarrow \left|A\right|\ne 0 \\ \Rightarrow \left|A\right|=k-29\ne 0 \\ \Rightarrow k\ne 29 \end{gathered}$$

Since the matrix A is invertible, then the vectors $\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 4\\ \mathrm{K}\end{array}\right]$ form a basis of ${\mathrm{ℝ}}^4{\mathrm{ℝ}}^4{\mathrm{ℝ}}^4{\mathrm{ℝ}}^4{\mathrm{ℝ}}^4$for all values of $k\in \mathrm{ℝ}-\left\{29\right\}$ .

Final Answer

The given vectors$\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 4\\ \mathrm{K}\end{array}\right]$ form a basis of ${\mathrm{ℝ}}^4$for all values of$k\in \mathrm{ℝ}\left\{29\right\}$