Q15E 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 5: Q15E (page 261)

For which values of the constants b, c, and dis the following an inner product in $R^{2}$ ?
$< [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] > = x_{1} y_{1} + b x_{1} y_{2} + c x_{2} y_{1} + d x_{2} y_{2}$
Hint: Be prepared to complete a square.

Short Answer

Expert verified

b = c and $d > b^{2}$

Step by step solution

Find constant a,b and c.

Here it needs to find out the constant a,b and c such that the following will be an inner product.

$< [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] > = x_{1} y_{1} + b x_{1} y_{2} + c x_{2} y_{1} + d x_{2} y_{2}$

Observe the following.

Let $[\begin{matrix} 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \end{matrix}] = b$ and $[\begin{matrix} 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 0 \end{matrix}] = c$

$鈬� b = c$

Now, the inner product is found as:

$[\begin{matrix} x_{1} \\ x_{2} \end{matrix}], [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = x_{1}^{2} + b x_{1} x_{2} + c x_{1} x_{2} + d x_{2}^{2} = x_{1}^{2} + b x_{1} x_{2} + b x_{1} x_{2} + d x_{2}^{2} = x_{1}^{2} + 2 b x_{1} x_{2} + d x_{2}^{2} = x_{1}^{2} + 2 (x_{1} + b x_{2}) + b^{2} x_{2}^{2} - b^{2} x_{2}^{2} + d x_{2}^{2} = (x_{1} + b x_{2}) + (d - b^{2}) x_{2}^{2}$

Being an inner product the last expression must be positive. For that we need $d > b^{2}$ .

Hence, for the following values is an inner product $b = c$ and $d > b^{2}$ .

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91影视

For which values of the constants b, c, and dis the following an inner product in $R^{2}$ ?
$< [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] > = x_{1} y_{1} + b x_{1} y_{2} + c x_{2} y_{1} + d x_{2} y_{2}$
Hint: Be prepared to complete a square.

Short Answer

Step by step solution

Find constant a,b and c.

One App. One Place for Learning.

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91影视

For which values of the constants b, c, and dis the following an inner product inR2 ?<x1x2,y1y2>=x1y1+bx1y2+cx2y1+dx2y2Hint: Be prepared to complete a square.

Short Answer

Step by step solution

Find constant a,b and c.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

For which values of the constants b, c, and dis the following an inner product in $R^{2}$ ?
$< [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] > = x_{1} y_{1} + b x_{1} y_{2} + c x_{2} y_{1} + d x_{2} y_{2}$
Hint: Be prepared to complete a square.