#flo #hw
1 Linear Maps
no one get's excited about vector spaces -axler
the interesting part: linear maps!
title: learning objectives
- fundementals theorem of linear maps
- matrix of linear map w.r.t. given bases
- isomorphic vec spaces
- product spaces
- quotient spaces
- duals spaces
- vector space
- linear map
2 The vector space of linear maps
key definition!
title: linear map aka *linear transformation.* a *linear map* from $V$ to $W$ is a function $T:V \to W$ with the following properties: **additivity** $T(u+v) = Tu+Tv$ for all $u, v \in V$; **homogeneity** $T(\lambda v) = \lambda(Tv)$ for all $\lambda \in F$ and $v \in V$.
the functional notation T(V) is the same as the notation Tv when talking about linear maps.
2.0.1 examples of linear maps
- 0?
- 0 is the func that takes each ele from some vec space to the
additive iden of another vec space.
- 0v = 0
- left: func from V to W, right: additive iden in W
- #question what does it mean for it to be a function from V to W?
- 0 is the func that takes each ele from some vec space to the
additive iden of another vec space.
- identity, denoted \(I\)
- \(Iv = v\)
- maps each element to itself linear transformation like a
.map?
- differentiation and integration!
- multiplication by \(x^2\) (on polynomials)
- shifts! defined as, \(T(x_1, x_2, x_3, \dots) = (x_2, x_3, \dots)\)
- #question this is an example, but how do we define it as a transformation? or is giving an example in the general case the same thing as defining a transformation?
- from \(R^3 \to R^2\) ? #question what? how does this work?
- #review how this dimension shift works..
- linear maps and basis of domain
title: linear maps and basis of domain Suppose $v_1, \dots , v_n$ is a basis of $V$ and $w_1, \dots , w_n \in W$. Then there exists a unique linear map $T:V \to W$ such that $$Tv_j = W_j$$ for each $j=1,\dots n$.
we can uniquely map between the basis of a subspace and a list of equal len in a diff subspace?
#question wait how does the uniqness proof work here at the end?
2.0.2 algebraic operations on \(L(V,W)\)
title: addition and SCAMUL Suppose $S,T \in L(V,W)$ and $\lambda \in F$. The *sum* of $S+T$ and the *product* $\lambda T$ are the linear maps from $V$ to $W$ defined by $$(S+T)(v) = Sv + Tv$$ and $$(\lambda T)(v) = \lambda (Tv)$$ for all $v \in V$
oh jeez..
title: $L(V,W)$ is a vector space! with the operations of addition and SCAMUL as defined aboce, $L(V,W)$ is a [[file:KBe20math530refVectorSpace.org][KBe20math530refVectorSpace]]
and another one.
- product of linear maps
title: product of linear maps if $T \in L(U,V)$ and $S \in L(V,W)$, then the *product* $ST \in L(U,W)$ is defined by $$(ST)(u)=S(Tu)$$ for all $u \in U$.
S dot T?? what is this symbol? it's a composition sign!!
\circ-> \(\circ\)title: albraic props of products of linear maps - associative - idenity - distributive properties
multiplication of linear maps is not commutative! ie. \(ST = TS\) isn't always true.
title: linear maps take 0 to 0 suppose $T$ is a linear map from $V$ to $W$. Then $T(0) = 0$
#review this chapter... bassically all just result blocks and nothing else i don't have an intuitive understanding of the concept of a map. perhaps look into 3b1b vid on linear transformations, or maybe professor dave.