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Q28E

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Found in: Page 143

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

For which value(s) of the constant k do the vectors below form a basis of ${{\mathbf{ℝ}}}^{{\mathbf{4}}}$ ?$\left[\begin{array}{c}1\\ 0\\ 0\\ 2\end{array}\right]\mathbf{}\mathbf{,}\mathbf{}\left[\begin{array}{c}0\\ 1\\ 0\\ 3\end{array}\right]\mathbf{}\mathbf{}\mathbf{,}\left[\begin{array}{c}0\\ 0\\ 1\\ 4\end{array}\right]\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\left[\begin{array}{c}2\\ 3\\ 4\\ K\end{array}\right]$

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 $k\in \mathrm{ℝ}-\left\{29\right\}$ .

See the step by step solution

Step 1: Definition of the basis

The vectors$\stackrel{\to }{{v}_{1}},\stackrel{\to }{{v}_{2}},...,\stackrel{\to }{{v}_{n}}$ 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

$A=\left[\stackrel{\to }{{v}_{1}}....\stackrel{\to }{{v}_{n}}\right]$ is invertible.

Step 2: Finding the determinant

Here we have

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

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

Step 3: 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.

$⇒\left|A\right|\ne 0\phantom{\rule{0ex}{0ex}}⇒\left|A\right|=k-29\ne 0\phantom{\rule{0ex}{0ex}}⇒k\ne 29$

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 $k\in \mathrm{ℝ}-\left\{29\right\}$ .

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 $k\in \mathrm{ℝ}\left\{29\right\}$