Q20E Consider the inner product(vr,wr... [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: Q20E (page 261)

Consider the inner product
$(\overset{r}{v}, \overset{r}{w}) = \overset{r}{v^{T}} [\begin{matrix} 1 & 2 \\ 2 & 8 \end{matrix}] \overset{r}{w}$ in $R^{2}$ . See Exercise 19.
a.Find all vectors in $R^{2}$ that are perpendicular to $[\begin{matrix} 1 \\ 0 \end{matrix}]$ with respect to this inner product.
b.Find an orthonormal basis of $R^{2}$ with respect to this inner product.

Short Answer

Expert verified

(a) The perpendicular vector is $[\begin{matrix} - 2 \\ 1 \end{matrix}]$ .

(b) The set will be span $[\begin{matrix} - 2 \\ 1 \end{matrix}]$ .

Step by step solution

Consider for part (a).

(a)

To find the all vectors which are perpendicular to $[\begin{matrix} 1 \\ 0 \end{matrix}]$ consider the solution

The set is found as:

$A = \{[\begin{matrix} x \\ y \end{matrix}] : [\begin{matrix} x \\ y \end{matrix}] 鈯� [\begin{matrix} 1 \\ 0 \end{matrix}]\} = \{[\begin{matrix} x \\ y \end{matrix}] : ([\begin{matrix} x \\ y \end{matrix}] 鈯� [\begin{matrix} 1 \\ 0 \end{matrix}]) = 0\} = \{[\begin{matrix} x \\ y \end{matrix}] : [x Y] [\begin{matrix} 1 & 2 \\ 4 & 8 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}]\} = \{[\begin{matrix} - 2 y \\ y \end{matrix}] : y 鈭� R\} = \{[\begin{matrix} - 2 \\ 1 \end{matrix}]\}$

Hence, the perpendicular vector is $[\begin{matrix} - 2 \\ 1 \end{matrix}]$ .

Consider for part (b).

(b)

From the part (a) the vectors $[\begin{matrix} 1 \\ 0 \end{matrix}]$ and $[\begin{matrix} - 2 \\ 1 \end{matrix}]$ are orthogonal, it only needs to make them unit length. Since,

$l e n g t h = ||[\begin{matrix} - 2 \\ 1 \end{matrix}]|| = \sqrt{4} = 2$

The orthonormal basis is $\{[\begin{matrix} - 2 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 0 \end{matrix}]\}$ .

Hence, the set will be span $[\begin{matrix} - 2 \\ 1 \end{matrix}]$ .

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Short Answer

Step by step solution

Consider for part (a).

Consider for part (b).

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91影视

Consider the inner product(vr,wr)=vTr[1228]wrin R2. See Exercise 19.a.Find all vectors inR2 that are perpendicular to[10] with respect to this inner product.b.Find an orthonormal basis ofR2 with respect to this inner product.

Short Answer

Step by step solution

Consider for part (a).

Consider for part (b).

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.