This is incredibly confusing. I also don't like the dot-product explanation for ...

nockupe_2 · on Jan 18, 2022

Totally yes. The mechanistic illustrations of matrix multiplication, including the OP, are easy enough but don't help me with motivation or intuition.

I always start with Ax, which is just a linear combination of the columns, like the first figure here https://eli.thegreenplace.net/2015/visualizing-matrix-multip...

For those who want to make their own cheat sheet:

Matrix $\times$ vector is linear combination of the columns:

\begin{align}

    Ax &= \begin{pmatrix}

    a_{1,1} & a_{1,2}\\

    a_{2,1} & a_{2,2}\\

    a_{3,1} & a_{3,2}

    \end{pmatrix}

\begin{pmatrix}
x_1\\x_2
\end{pmatrix}\\
&= x_1\begin{pmatrix}
a_{1,1}\\
a_{2,1}\\
a_{3,1}\\
\end{pmatrix}
+
x_2\begin{pmatrix}
a_{1,2}\\
a_{2,2}\\
a_{3,2}\\
\end{pmatrix}

\end{align}

Intuitively, this example maps a point in $\mathbb{R}^2$ to $\mathbb{R}^3$

Another way to see $Ax$: Columns of $A$ form a basis, and $x$ is a coordinate vector in that basis.

vecter · on Jan 19, 2022

The link you posted perfectly depicts what I was saying. It's good to see a clear example of that online, when almost every other resource is showing the dot-product interpretation.

commandlinefan · on Jan 18, 2022

> a linear combination of column vectors

Isn’t that pretty much the same as the dot-product explanation?

vecter · on Jan 18, 2022

To clarify, I mean if the columns of a 3x3 matrix A are A1, A2, and A3, and the scalar elements of vector x are <x1, x2, x3> then Ax = x1*A1 + x2*A2 + x3*A3. Each column of A is scaled by an element of x and then added together.

Is that what you had in mind by the dot-product explanation? To me, the dot product explanation is that in Ax = b, b1 = <row1 of A> dot x, b2 = <row2 of A> dot x, and b3 = <row3 of A> dot x.

Of course these (and all other valid) interpretations of matrix multiplication are "the same", but this is less geometrically intuitive to me.