Q27E Question: Consider an n脳mmatrix... [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: Q27E (page 233)

Question: Consider an $n 脳 m$ matrix A, a vector in $R^{m}$ , and a vector $\overset{鈫�}{w}$ in $R^{m}$ . Show that $(A \overset{鈫�}{v}) . \overset{鈫�}{w} = \overset{鈫�}{v} . (A^{T} \overset{鈫�}{w})$ .

Short Answer

Expert verified

It is proved that $(A \overset{鈫�}{v}) . \overset{鈫�}{w} = \overset{鈫�}{v} . (A^{T} \overset{鈫�}{w})$ .

Step by step solution

Consider the theorem

If $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two (column) vectors in $R^{n}$ , then $\begin{matrix} \overset{\overset{鈫�}{v} . \overset{鈫�}{w}}{鈫�} & \overset{= {\overset{鈫�}{v}}^{T} \overset{鈫�}{v}}{鈫�} \\ \frac{(dot) (matrix)}{product} \end{matrix}$ .

Application of the theorem

Observe that,

$(A \overset{鈫�}{v}) . \overset{鈫�}{w} = {(A \overset{鈫�}{v})}^{T} \overset{鈫�}{w} (t h e o r e m 5.3.6) = ({\overset{鈫�}{v}}^{T} A^{T}) \overset{鈫�}{w (t h e o r e m 5.3.9 (c))} = {\overset{鈫�}{v}}^{T} (A^{T} \overset{鈫�}{w}) (A s s o c i a t i v i t y) = \overset{鈫�}{v} . ((A^{T} \overset{鈫�}{w}))$ According to the theorem.

Hence, $(A \overset{鈫�}{v}) . \overset{鈫�}{w} = \overset{鈫�}{v} (A^{T} \overset{鈫�}{w})$ is proved.

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91影视

Question: Consider an $n 脳 m$ matrix A, a vector in $R^{m}$ , and a vector $\overset{鈫�}{w}$ in $R^{m}$ . Show that $(A \overset{鈫�}{v}) . \overset{鈫�}{w} = \overset{鈫�}{v} . (A^{T} \overset{鈫�}{w})$ .

Short Answer

Step by step solution

Consider the theorem

Application of the theorem

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91影视

Question: Consider an n脳mmatrix A, a vector in Rm, and a vector w鈫�in Rm. Show that (Av鈫�).w鈫�=v鈫�.(ATw鈫�).

Short Answer

Step by step solution

Consider the theorem

Application of the theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Geometry

Calculus

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: Consider an $n 脳 m$ matrix A, a vector in $R^{m}$ , and a vector $\overset{鈫�}{w}$ in $R^{m}$ . Show that $(A \overset{鈫�}{v}) . \overset{鈫�}{w} = \overset{鈫�}{v} . (A^{T} \overset{鈫�}{w})$ .