- #1
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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