Q 5.5-7E Question: Consider an inner prod... [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: Q 5.5-7E (page 260)

Question: Consider an inner product $鉄� v, w 鉄�$ in a space V, and a scalar k. For which choices of kis
$鈱� 鈱�v, w鈱� 鈱� = k 鈱� v, w 鈱�$ an inner product?

Short Answer

Expert verified

$鉄� 鉄�, 鉄� 鉄�$ is an inner product.

Step by step solution

Find out k.

Here V is an inner product space and for $v, w 鈭� V$ ,k is scalar then,

It wants to find out all k such that the above will also be an inner product.

Observe that for $v, w 鈭� V$ and c any scalar.

$\begin{array}{l} 鈱�鈱�u, v鈱�鈱� = k 鈱�u, v鈱� \\ = k 鈱�v, u鈱� \\ = 鈱�鈱�v, u鈱�鈱� \end{array}$
$\begin{array}{l} 鈱�鈱�u + v, w鈱�鈱� = k 鈱�u + v, w鈱� \\ = k 鈱�鈱�u, w鈱� + 鈱�v, w鈱�鈱� \\ = k 鈱�u, w鈱� + k 鈱�v, w鈱� \\ = 鈱�鈱�u, w鈱� + 鈱�v, w鈱�鈱� \end{array}$
$\begin{array}{l} 鈱�鈱�c u, v鈱�鈱� = k 鈱�c u, v鈱� \\ = k (c 鈱�u, v鈱�) \\ = c (k 鈱�u, v鈱�) \\ = c 鈱�鈱�c u, v鈱�鈱� \end{array}$

Let $u 鈮� 0$

$鈱�鈱�u, u鈱�鈱� = k 鈱�u, u鈱� 鈮� 0 鈬� k 鈮� 0$ and $鈱�u, u鈱� 鈮� 0$

Since $u 鈮� 0$ then $鈱�u, u鈱� 鈮� 0$ and k has to be non-zero.

Hence, $鈱�鈱�,鈱�鈱�$ is an inner product if $k 鈭� R \ {0}$ .

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91影视

Question: Consider an inner product $鉄� v, w 鉄�$ in a space V, and a scalar k. For which choices of kis
$鈱� 鈱�v, w鈱� 鈱� = k 鈱� v, w 鈱�$ an inner product?

Short Answer

Step by step solution

Find out k.

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91影视

Question: Consider an inner product 鉄�v,w鉄�in a space V, and a scalar k. For which choices of kis鈱�v,w鈱�=k鈱�v,w鈱�an inner product?

Short Answer

Step by step solution

Find out k.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Decision Maths

Mechanics Maths

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: Consider an inner product $鉄� v, w 鉄�$ in a space V, and a scalar k. For which choices of kis
$鈱� 鈱�v, w鈱� 鈱� = k 鈱� v, w 鈱�$ an inner product?