Table of Contents
#flo #inclass
1 Linear maps!
when we hear linear, we should think: addition and scalar multiplication!
if it's in a box you should know it! -jana
homogenity: works nicely under SCAMUL - what does nicely mean? - addition and mapping is commutative? - can add, then send through map, which = send through map then add
- MULTI first then map = map then MULTI
this is called: homomorphism
title: homomorphism structure-preserving map between two algebraic structures of the same type. -wiki
1.0.1 examples!
KBxChapter3AReading#examples of linear maps
- how \(F^n\) to \(F^m\) works
- inp: nx1 mat
- oup: mx1 mat
- which means: the org thing needs to be mxn mat
- 3x1 -> 2x1
$
\begin{bmatrix} 2 & -1 & 3 \\ 7 & 5 & -6 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix}=
\begin{bmatrix} 2x-y+3z \\ 7x+5y-6z \end{bmatrix}$ uhh..
- backwards shift!
- [0 1 0 0 0]
- [0 0 1 0 0]
- this just drops the first element, and is essentially a repeat of above example of mapping to a lower dimension
- #question wait how do u actully define this operation? just create an example with arbitrary elems?
if you know where the basis goes, then you know where everything else goes! @3b1b talked about this KBxChapter3AReading#linear maps and basis of domain
1.0.2 algebraic operations
this is how we combine maps!
- how we are used to adding functions
we get.. addition and SCAMUL which makes:
[[file:KBxL(V,W).org][KBxL(V,W)]] a [[file:KBe20math530refVectorSpace.org][KBe20math530refVectorSpace]]!
inp: nx1, oup: mx1 for KBxChapter3AReading#product of linear maps
if the domain of one matches the KBxCodomain of another