Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the basis

The vectors $\vec{v_{1}}, \vec{v_{2}}, . . ., \vec{v_{n}}$ in $ℝ^{n} ℝ^{n}$ form a basis of $ℝ^{n} ℝ^{n} ℝ^{n}$ if (and only if) the matrix

$A = [\vec{v_{1}} . . . . \vec{v_{n}}]$ is invertible.

Step 2: Finding the determinant

Here we have

$A = [\begin{matrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 2 & 3 & 4 & k \end{matrix}]$

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

Step 3: Finding the values of k

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

$\Rightarrow |A| \neq 0 \Rightarrow |A| = k - 29 \neq 0 \Rightarrow k \neq 29$

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

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of the basis

Step 2: Finding the determinant

Step 3: Finding the values of k

Step 4: Final Answer

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Linear Algebra With Applications

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Short Answer

For which value(s) of the constant k do the vectors below form a basis of ℝ4 ?[1002] , [0103] ,[0014] , [234K]

Step by Step Solution

Step 1: Definition of the basis

Step 2: Finding the determinant

Step 3: Finding the values of k

Step 4: Final Answer

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