Show that if u and v are each n X 1column vectors, then

Chapter 9, Problem 7E

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Show that if $u \text { and } v$ are each $n \times 1$ column vectors, then the matrix product $u^{T} v$ is the same as the dot product $u \cdot v$

Let v be a $3 \times 1$ column vector with v^{T}=\left[\begin{array}{lll} $2 & 3 & 5 \end{array}\right]$

Show that, for B as given in Example 1, $(A v)^{T}=v^{T} A^{T}$

Does $(A v)^{T}=v^{T} A^{T}$ hold for every $m x n $ matrix $A$ and $n \times 1$ vector $v$?

Does $(A B)^{T}=B^{T} A^{T}$ hold for every pair of matrices $A, B$ such that both matrix products are defined? Justify your answer.

Equation Transcription:

$u\ and\ v$

$n\ x\ 1$

${u}^{T}v$

$u\cdot{}v$

$3\ x\ 1$

${v}^{T}=[2\ \ 3\ \ 5]$

$(Av{)}^{T}={v}^{T}{A}^{T}$

$m\ x\ n$

$A$

$n\ x\ 1$

$v$

$(AB{)}^{T}={B}^{T}{A}^{T}$

$A,B$

Text Transcription:

u and v

n x 1

uTv

3 x 1

vT=[2 3 5]

(Av)T=vTAT

m x n

n x 1

(AB)T=BTAT

A,B

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