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This is incredibly confusing. I also don't like the dot-product explanation for individual entries of the resultant matrix. When I was in college, it was hammer

by vecter 5y ago

This is incredibly confusing. I also don't like the dot-product explanation for individual entries of the resultant matrix. When I was in college, it was hammered into my head that matrix multiplication was a linear combination of column vectors. It took a while for that to sink in, but once it did, it made a lot more geometric sense than just taking a bunch of inner products of column and row vectors.

commandlinefan 5y ago
> a linear combination of column vectors Isn’t that pretty much the same as the dot-product explanation?
- vecter 5y ago
  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.
nockupe_2 5y ago
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-multiplication-as-a-linear-combination/ 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 5y ago
  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.

by vecter 5y ago

commandlinefan 5y ago

> a linear combination of column vectors Isn’t that pretty much the same as the dot-product explanation?

vecter 5y ago
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.

nockupe_2 5y ago

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-multiplication-as-a-linear-combination/ 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 5y ago
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.