0
\$\begingroup\$

I know the bare minimum to do matrix multiplication and have implemented them from scratch a few times for various purposes, but every time I do, I make sure to be very clear to myself about where are the basis vectors.

I dont think of matrices as big squares of numbers, but as arrays of vectors where each component is.. well, a basis component.

But as I have learned about matrices through alternate means, I have no idea of how this commonly used notation works, and if there's an implicit common sense way to distribute the basis vectors.

people usually write matrix multiplication in the following notation:
enter image description here

but its very ambiguous to me how to interpret it. my brain is drawn by the visual layout to interpret it as the following:

enter image description here

but most people seem to actually mean this:

enter image description here

so which is the correct way to interpret it?

\$\endgroup\$

1 Answer 1

Reset to default
0
\$\begingroup\$

Neither of the pictures you've shown corresponds to matrix multiplication, in which each row of the matrix is multiplied (as a dot product) with the full column of the input vector to produce one component of the output vector.

Your modified second drawing is closer, where you can think of the output as a linear combination of the columns of the matrix, with the components of the input forming the weights.

If you're wondering how to interpret the rows or columns of the matrix, then in the form you've shown, each column of the upper-left 3x3 submatrix is a basis vector of the destination space. So the left column is the image of (1, 0, 0) after multiplication, the middle column is the image of (0, 1, 0), etc. The fourth column is the translation: the image of (0, 0, 0).

I explain this interpretation in more detail in this older answer.

\$\endgroup\$
4
  • \$\begingroup\$ I think of matrix multiplication as a linear combination of basis vectors, where the vector being transformed is an array of weights. Sums of dot products are confusing to me, so thats the source of my confusion. \$\endgroup\$ Commented Apr 4, 2022 at 2:33
  • \$\begingroup\$ These interpretations are equivalent, but then the circles around your output vector should encompass the whole vector, not just one component, since each basis vector contributes to all four components. \$\endgroup\$ Commented Apr 4, 2022 at 2:38
  • \$\begingroup\$ yea, my mistake, each circle in my drawing is supposed to mean one scalar product of one component of the vector by one basis vector of the matrix, so the final vector is a sum of all those scalar products. \$\endgroup\$ Commented Apr 4, 2022 at 2:42
  • \$\begingroup\$ I fixed the drawings \$\endgroup\$ Commented Apr 4, 2022 at 2:55

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.