Q3.2-56E For which values of the constant... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.2-56E (page 133)

For which values of the constants $a, b, . . . . ., m$ are the given vectors linearly independent?
$[\begin{matrix} a \\ b \\ c \\ d \\ 1 \\ 0 \end{matrix}], [\begin{matrix} e \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} f \\ g \\ h \\ i \\ j \\ l \end{matrix}], [\begin{matrix} k \\ m \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$

Short Answer

Expert verified

For any values of $a, b, . . . . ., m$ the vectors,localid="1664209238077" $[\begin{matrix} a \\ b \\ c \\ d \\ 1 \\ 0 \end{matrix}], [\begin{matrix} e \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} f \\ g \\ h \\ i \\ j \\ l \end{matrix}], [\begin{matrix} k \\ m \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ can be linearly independent.

Step by step solution

Consider the set

The vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, 鈥�, {\overset{鈫�}{v}}_{m}$ in ${鈩�}^{n}$ are linearly dependent if and only if there are nontrivial relation among them.

If linearly dependent, then, the vectors are said to be redundant.

Consider the vector role="math" localid="1664209158244" $[\begin{matrix} a \\ b \\ c \\ d \\ 1 \\ 0 \end{matrix}], [\begin{matrix} e \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} f \\ g \\ h \\ i \\ j \\ l \end{matrix}], [\begin{matrix} k \\ m \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$

Consider the condition

For the vectors, ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4}$

$c_{1} {\overset{鈫�}{v}}_{1} + c_{2} {\overset{鈫�}{v}}_{2} + c_{3} {\overset{鈫�}{v}}_{3} + c_{4} {\overset{鈫�}{v}}_{4} = 0$

The solution will be trivial for any values of $a, b, . . . . ., m$ .

Final answer

For any values of $a, b, . . . . ., m$ the vectors, $[\begin{matrix} a \\ b \\ c \\ d \\ 1 \\ 0 \end{matrix}], [\begin{matrix} e \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} f \\ g \\ h \\ i \\ j \\ l \end{matrix}], [\begin{matrix} k \\ m \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ can be linearly independent.

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91影视

Short Answer

Step by step solution

Consider the set

Consider the condition

Final answer

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91影视

For which values of the constantsa,b,.....,m are the given vectors linearly independent?[abcd10],[e10000],[fghijl],[km1000]

Short Answer

Step by step solution

Consider the set

Consider the condition

Final answer

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Most popular questions from this chapter

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