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TheSodesa

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## Homework Statement

Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation,

[tex]

B = \begin{bmatrix}

b_1 , & b_2, & ... & ,b_m

\end{bmatrix}

[/tex]

Prove that

[tex]AB =

\begin{bmatrix}

Ab_1 , & Ab_2, & ... & , Ab_m

\end{bmatrix}

[/tex]

If ##A## is represented with help of its row vectors, prove that

[tex]

AB =

\begin{bmatrix}

a^{T}_{1}\\

\vdots\\

a_{n}^{T}

\end{bmatrix} B

=

\begin{bmatrix}

a^{T}_{1} B\\

\vdots\\

a^{T}_{n} B

\end{bmatrix}

[/tex]

## Homework Equations

The matrix product:

If ##A## is an ##m\times p## matrix and ##B## is a ##p\times n## matrix, then

\begin{equation}

AB = C = (c_{ij})_{m \times n} = (\sum_{k=1}^{p} a_{ik}b_{kj})_{m \times n}

\end{equation}

## The Attempt at a Solution

For starters what does proving in this context mean? Should I simply write out the matrix

[tex]

C =

\begin{bmatrix}

\sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\

\vdots & \ddots & \vdots\\

\sum_{k=1}^{p} a_{mk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km}

\end{bmatrix}

[/tex]

and conclude that each column is essentially ##Ab_{l}## where ##1 < l < m##, since each element of the matrix is the dot (inner) product of a row in ##A## and a column in ##B##?

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