Backlinks
1 Looking forward
- Will use canvas's discussion board in the future.
- Assume matrices have real numbers
2 Solving with Matrices
- Elementary matrices (like \(\left[\begin{matrix}1 &-2 \\ 0 &1\end{matrix}\right]\))
- Steps walk through
- Start with \(\left[\begin{matrix}a&b\\d&e\end{matrix}\right]\) (the coefficient matrix).
- You want to get somewhere such that \(\left[\begin{matrix}1x\\0y\end{matrix}\right] = \left[\begin{matrix}c\\f\end{matrix}\right]\)
- And ultimately \(\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left[\begin{matrix}x\\y\end{matrix}\right]=\left[\begin{matrix}{ans}_x\\{ans}_y\end{matrix}\right]\)
- srcD3SolveWithMatricies.png
3 Matrix Inverse Formula
- I should technically know this already.
3.1 Derivation
\[
\left[\begin{matrix}a&b\\c&d\end{matrix}\right]
\left[\begin{matrix}w&x\\y&z\end{matrix}\right]
\left[\begin{matrix}aw+by&ax+bz \\ cw+dy&cx+dz\end{matrix}\right]
\\∴
\]
- There's two 2 variable equations. srcIdentityMatrixFormula.png
4 Matrix Operations
- If we have a set of objects that are almost groups in under both
addition and multiplication, then it's called a field
- 2x2 Matrices aren't quite close enough on the multiplication (too many no inverses) but we can work with other sizes. ### Vector Products
- Matrices of dimension \(n\)x\(1\)
- What multiplications on vectors are "nice"?
- Transpose the first (left) one and multiply normally, then squish 2x2 into 2x1
- Cross product
- Element wise (is closed)
- Take every element and multiply them all together, and then
duplicate?
- No, no identity
- Any one to one mapping?
- No, identity doesn't work if it's on the left.