What sizes of matrix can you add? When can't you add matrices?
Matrices of the same dimensions (because we do it element wise). Maybe you can add a vector to a matrix if the number of rows is equal to the dimensionality of the vector.
What sizes of matrix can you multiply? When can't you multiply matrices?
Multiply: \(N\times M\) * \(M\times K\) => \(N\times K\).
Multiply \[
\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\] by vectors in \(\mathbb{R}^2\) (for example, you could multiply by \(\begin{bmatrix} 0\\ 0 \end{bmatrix}\) or \(\begin{bmatrix} 1\\ -2 \end{bmatrix}\)).
Can you characterize the transformations you get by multiplying (lots of vectors) by each of these matrices?
| Action | Matrix |
|---|---|
| Identity | \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\) |
| Select left column | \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) |
| Select right column | \(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\) |
| Treat as expression (linear combination/transformation?)* | \(\begin{bmatrix} a \\ b \end{bmatrix}\) |
*I'm not sure what linear combinations/transformations are, but I think this is somehow related? Anyways, it takes each row \(i\) and returns \(\sigma A_{i,j} * B_{j}\)
Which of the number systems we discussed today form a group under addition? Under multiplication?
Source: KBe2020math530refGroups
| Number System | Multiplication | Addition |
|---|---|---|
| Natural Numbers | No inverse | No identity |
| Whole Numbers | No inverse | No inverse |
| Integers | No inverse | Yes |
| Rationals | Yes | Yes |
| Reals | Yes | Yes |
| Complex Numbers | Yes | Yes |