Q28E For which value(s) of the consta... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q28E (page 143)

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

Short Answer

Expert verified

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 ${鈩�}^{4} {鈩�}^{4} {鈩�}^{4}$ for all values of $k 鈭� 鈩� - \{29\}$ .

Step by step solution

Definition of the basis

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

$A = [\overset{鈫�}{v_{1}} . . . . \overset{鈫�}{v_{n}}]$ is invertible.

Finding the determinant

Here we have

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

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

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 ${鈩�}^{4}$ 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.

$鈬� |A| 鈮� 0 鈬� |A| = k - 29 鈮� 0 鈬� k 鈮� 29$

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 ${鈩�}^{4} {鈩�}^{4} {鈩�}^{4} {鈩�}^{4} {鈩�}^{4}$ for all values of $k 鈭� 鈩� - \{29\}$ .

Final Answer

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91影视

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

Short Answer

Step by step solution

Definition of the basis

Finding the determinant

Finding the values of k

Final Answer

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91影视

For which value(s) of the constant k do the vectors below form a basis of 鈩�4?[1002],[0103],[0014],[234K]

Short Answer

Step by step solution

Definition of the basis

Finding the determinant

Finding the values of k

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

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