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What is the geometry behind the matrix multiplication?

The questions that I am having is the follows.

$\bullet$ I accept that we are viewing $\mathbb R^4$ as either in $\begin{pmatrix} a_{11},a_{12},a_{13},a_{14}\end{pmatrix}$ or $\begin{pmatrix} a_{11}&a_{12}\\a_{13}&a_{14}\end{pmatrix}$. so, matrix addition makes sense that it gives another vector in that space.

$\bullet$ But, Matrix multiplication does not convince me in this role.

I strucked with,

Like in the matrix addition, (addition of two vectors is nothing but the diagonal of the parallelogram in which the two vectors are adjustcent sides)

is there any vector space diagramatic representation for matrix multiplication??

Note: I am aiming to teach this factacy to my grade 11 students who are studying their matrices now only.(Means to say, this is the first time they gonna meet matrices)

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  • $\begingroup$ Geometrically, matrix multiplication corresponds to composition of linear transformations. $\endgroup$ – jkabrg Jun 24 '15 at 17:59
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    $\begingroup$ Factasy? Sounds like it should be a word... "rapturous delight caused by learning a new fact". $\endgroup$ – Robert Israel Jun 24 '15 at 22:47
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Another perspective is to use the SVD, hence a matrix multiplication can be thought of as a rotation, followed by a (per dimension) scaling, followed by another rotation.

That is any (real) matrix $A$ can be written as $A=U \Sigma V^T$, where $U,V$ are rotations and $\Sigma$ is diagonal, with non negative entries (in decreasing order on the diagonal).

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  • $\begingroup$ May you give me an example which explains your answer!!!!!@copper $\endgroup$ – David Jun 26 '15 at 17:12
  • $\begingroup$ because your answer seems to be better and easy to explain to the students..@copper $\endgroup$ – David Jun 26 '15 at 17:13
  • $\begingroup$ @gloom: On reflection I'm not so sure that the above a good approach. I was thinking of matrix multiplication as in multiplying a single vector as opposed to another matrix. Obviously matrix multiplication can be interpreted as multiplying the columns, but I don't think this will clarify anything for 11th graders. $\endgroup$ – copper.hat Jun 26 '15 at 17:53
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The "geometry" behind matrix multiplication is the composition of the corresponding linear maps. The matrices $A\in M(n\times m, \mathbb k)$ and $B \in M(m \times p, \mathbb k)$ define with respect to the canonical bases linear maps $f_B: \mathbb k^p \longrightarrow \ \mathbb k^m$ and $f_A: \mathbb k^m \longrightarrow \ \mathbb k^n$. Similar, the product $C:=AB \in M(n\times p, \mathbb k)$ defines a linear map $f_C: \mathbb k^p \longrightarrow \ \mathbb k^n$. One proves that the linear map $f_C$ is the composition $f_C = f_A \circ f_B$. Here $\mathbb k$ denotes the base field, e.g., $\mathbb k = \mathbb R$.

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There are several points of view to interpret matrix multiplication. Let's say you're doing $A B = C$ where $A$ is $m \times n$ and $B$ is $n \times p$, so $C$ is $m \times p$.

  1. Each column of $C$ is the matrix-vector product of $A$ with the corresponding column of $B$. Thus the $j$'th column of $C$ is a linear combination of the columns of $A$ with coefficients given by the entries of the $j$'th column of $B$.
  2. Each row of $C$ is the vector-matrix product of the corresponding row of $A$ with the matrix $B$. Thus the $i$'th row of $C$ is a linear combination of the rows of $B$, with coefficients given by the entries of the $i$'th row of $A$.
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