3. Multiplication and Inverse Matrices

What people said

Recommends saulspatz · 2018-10 · Mathematics Stack Exchange · on “Can I describe a matrix multiplication as a transformation of one matrix by another?”

Yes, there are certainly ways to see this. Gilbert Strang discusses various interpretations of matrix multiplication in this lecture [this video] Well, I'm pretty sure it's this lecture, but I haven't watched the videos recently. I think you got a little confused with the or order of operations. It seems like you are computing $ABx,$ so you would multiply $x$ by $B$ first, not by $A$ . If you want to consider $A$ as being transformed by $B$ , think of $A$ as $<A_1,A_2,\dots,A_n>$ where the $A_i$ are the columns of $A$ . Then it's easy to see that the columns of $BA$ are just $<BA_1, BA_2, \dots,BA_n>.$ That is, we just apply $B$ to the columns of $A$ . Other possible interpretations are given in

Recommends BeastCoder2 · 2019-07 · Mathematics Stack Exchange · on “Do all matrix multiplication methods boil down to dot product?”

Following the video [this video] from MIT's [LINK] OCW [LINK] , the Prof. brings up many different methods, like $row * matrix$ , $column * matrix$ , etc. Yet don't these all come down to dot product. For example $row * matrix$ would be done using dot product, right? Is the Prof just showing all of these things to show the big picture, or what? A little bonus: The Prof says that with the multiplication of the matrices A and B (AB) equaling to C, that the rows of C are a combination of the rows of A. Here, the columns of B must be multiplied by the matrix A (column * matrix method) to get one column of C. So, to do this you would end up doing the columns of B times all the rows of A (using dot pro

Recommends Tim · 2023-06 · Mathematics Stack Exchange · on “Understanding the proof of PCA”

for the matrix multiplication part, the link teach different types of matrix multiplication, you may refer to [this video]

Cites Will Clyne · 2013-04 · Mathematics Stack Exchange · on “Matrix Multiplication Interpretation”

According to Lecture 3 [this video] (from approximately 07:00 - 12:00) of Gilbert Strang's course on linear algebra, Given two matrices $A$, $B$ and their product $C$: 1) Columns of C are combinations of columns of A 2) Rows of C are combinations of rows of B However, to me, it seems the opposite is observed. My confusion is probably best illustrated explicitly. Suppose, as above, two matrices and their product such that: $$ \begin{bmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \ldots & a_{mn} \end{bmatrix} \times \begin{bmatrix} b_{11} & \ldots & b_{1p} \\ \vdots & \ddots & \vdots \\ b_{n1} & \ldots & b_{np} \end{bmatrix} = \begin{bmatrix} c_{11} & \ldots & c_{1p} \\ \