Q8E Find a nontrivial relation among... [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: Q8E (page 131)

Find a nontrivial relation among the following vectors:
$[\begin{matrix} 1 \\ 2 \end{matrix}], [\begin{matrix} 2 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 4 \end{matrix}]$ .

Short Answer

Expert verified

A non-trivial relation between the required vectors is, $[\begin{matrix} 3 \\ 4 \end{matrix}] = - [\begin{matrix} 1 \\ 2 \end{matrix}] + 2 |\begin{matrix} 2 \\ 3 \end{matrix}]$

Step by step solution

Consider the set.

A subset W of the vector space ${鈩�}^{n}$ is called a (linear) subspace of ${鈩�}^{n}$ if it has the following three properties:

a. W contains the zero vector in ${鈩�}^{n}$ .

b. W is closed under addition: If ${\overset{鈫�}{w}}_{1}$ and ${\overset{鈫�}{w}}_{2}$ are both in W, then so is ${\overset{鈫�}{w}}_{1} + {\overset{鈫�}{w}}_{2}$ .

c. W is closed under scalar multiplication: If $\overset{鈫�}{w}$ is in W and k is an arbitrary scalar, then $k \overset{鈫�}{w}$ is in W.

Let $v_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}], v_{2} = [\begin{matrix} 2 \\ 3 \end{matrix}], v_{3} = [\begin{matrix} 3 \\ 4 \end{matrix}]$ and $v_{3} = c_{1} v_{1} + c_{2} v_{2}$

Consider the matrix representation.

The matrix representation of the above vectors is,

$v_{3} = c_{1} v_{1} + c_{2} v_{2} 鈬� [\begin{matrix} 3 \\ 4 \end{matrix}] = c_{1} [\begin{matrix} 1 \\ 2 \end{matrix}] + c_{1} [\begin{matrix} 2 \\ 3 \end{matrix}] 鈬� [\begin{matrix} c_{1} + 2 c_{2} \\ 2 c_{1} + 3 c_{2} \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Solve the above equations.

$c_{1} = - 1, c_{2} = 2$

Final answer.

A non-trivial relation between the required vectors is, $[\begin{matrix} 3 \\ 4 \end{matrix}] = - [\begin{matrix} 1 \\ 2 \end{matrix}] + 2 |\begin{matrix} 2 \\ 3 \end{matrix}]$

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91影视

Find a nontrivial relation among the following vectors:
$[\begin{matrix} 1 \\ 2 \end{matrix}], [\begin{matrix} 2 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 4 \end{matrix}]$ .

Short Answer

Step by step solution

Consider the set.

Consider the matrix representation.

Final answer.

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91影视

Find a nontrivial relation among the following vectors: [12,23,34] .

Short Answer

Step by step solution

Consider the set.

Consider the matrix representation.

Final answer.

One App. One Place for Learning.

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Find a nontrivial relation among the following vectors:
$[\begin{matrix} 1 \\ 2 \end{matrix}], [\begin{matrix} 2 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 4 \end{matrix}]$ .