Q19E For which 2聽脳聽2 matrices A聽i... [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 5: Q19E (page 261)

For which 2脳2 matrices Ais
$(\overset{鈫�}{v}, \overset{鈫�}{w}) = {\overset{鈫�}{v}}^{T} A \overset{鈫�}{w}$ an inner product in $R^{2}$ ? Hint: Be prepared to complete a square.

Short Answer

Expert verified

The given product is an inner product if a>0, b=c and $d > b^{2}$

Step by step solution

Inner product of the matrix.

To find the inner product of the matrix.

$A = [\begin{matrix} a & b \\ c & d \end{matrix}], \overset{r}{v} = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}], \overset{r}{w} = [\begin{matrix} w_{1} \\ w_{2} \end{matrix}]$

Let ,

Consider the following:

$(\overset{r}{v}, \overset{r}{w}) = [\begin{matrix} v_{1} & v_{2} \end{matrix}] [\begin{matrix} a & b & w_{1} \\ c & d & w_{2} \end{matrix}] = [\begin{matrix} v_{1} & v_{2} \end{matrix}] [\begin{matrix} a w_{1} + b w_{2} \\ c w_{1} + d w_{2} \end{matrix}] = a w_{1} v_{1} + b w_{2} v_{2} + c w_{1} v_{2} + d w_{2} v_{2}$

Now, observe that if $\overset{r}{v} = [\begin{matrix} 1 \\ 0 \end{matrix}] = \overset{r}{w}$ then, $(\overset{r}{v}, \overset{r}{v}) = a > 0$ .

Thus, it is written as $(\overset{r}{v}, \overset{r}{v}) = a (w_{1} v_{1} + \frac{b}{a} w_{2} v_{2} + \frac{c}{a} w_{1} v_{2} + \frac{d}{a} v_{2} w_{2})$ .

But from the exercise 15 it will be an inner product if and only if when $\frac{b}{a} = \frac{c}{a}$ and $\frac{d}{a} > {(\frac{b}{a})}^{2}$ .

Therefore, a>0, b=a and $d > b^{2}$ .

Hence, the given product will be an inner product if a>0, b=c and $d > b^{2}$ .

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91影视

For which 2脳2 matrices Ais
$(\overset{鈫�}{v}, \overset{鈫�}{w}) = {\overset{鈫�}{v}}^{T} A \overset{鈫�}{w}$ an inner product in $R^{2}$ ? Hint: Be prepared to complete a square.

Short Answer

Step by step solution

Inner product of the matrix.

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91影视

For which 2脳2 matrices Ais(v鈫�,w鈫�)=v鈫�TAw鈫� an inner product in R2? Hint: Be prepared to complete a square.

Short Answer

Step by step solution

Inner product of the matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For which 2脳2 matrices Ais
$(\overset{鈫�}{v}, \overset{鈫�}{w}) = {\overset{鈫�}{v}}^{T} A \overset{鈫�}{w}$ an inner product in $R^{2}$ ? Hint: Be prepared to complete a square.