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#flo #hw
1 Null Spaces and Ranges
Two subspaces that are connected with every linear map
The set of vectors that get mapped to 0 is called the
title: null space, null T, AKA kernel
for $T \in$ [[file:KBxL(VcmW).org][KBxL(VcmW)]], the *null space* of $T$, denoted null $T$, is the subset of $V$ consisting of those vectors that $T$ maps to 0:
$$
null\ T = {v \in V : Tv = 0}.
$$
1.0.1 examples of null space
- The zero map! ie, \(Tv=0\) then \(\textrm{null}\ T = V\)
- \(D \in L(P(R), P(R))\) where \(Dp=p'\), then the constant funcs are
gonna go to 0.
- ie. null space of \(D\) equals the set of constant functions.
- backwards shift by one, \(\textrm{null}\ T = {(a, 0,0,...)}\)
- null space of each linear transformation is a subspace of the domain?
- ie, the kernel is a subspace.. oh boy
1.0.2 as a subspace
title: the null space is a subspace
Suppose $T \in L(V,W)$. Then $\textrm{null}\ T$ is a subspace of $V$
trivial proof, just plug in zeros.
ooh, and now we get,
1.0.3 injective
title: injective a function $T: V \to W$ is called *injective* if $Tu = Tv$ implies $u=v$
#question what does "implies" mean here?
he calls this, one-to-one, but this only works one way. else, it's bijective!
this means, that T is injective if it maps distinct inputs to distinct outputs
1.0.4 Range and Surjectivity
time to define, range!
title: range AKA image
for $T$ a function from $V$ to $W$, the *range* of $T$ is the subset of $W$ consisting of those vectors that are of the form of $Tv$ for some $v \in V$:
$$
\textrm{range} \ T = \{Tv: v \in V\}.
$$
just the… normal def of range.
and some examples: - are in 3.18
title: the range is a subspace! If $T \in L(V,W),$ then range $T$ is a subspace $W$.
and ofc,
title: surjective AKA onto a function $T: V \to W$ is called *surjective* if its range equals $W$.
surjecitvity depends on the space we are mapping into
1.0.5 Fundemental theorem
this is important! that's why the name is dramatic.
title: Fundemental theorem of linear maps Suppose $V$ is finite-dimensional $T \in L(V,W).$ Then range $T$ is finite-dimensional and $$\dim V = \dim null \ T + \dim range \ T$$
1.0.6
uh..
def of a smaller vec space is one with less a smaller dim
we can say that no linear transformation from a finite-dimensional vec space to a smaller vec space can be injective which makes sense! because you need the repeat elements, otherwise it woudnt be smaller.
title: A map to a smaller dimensinal space is not injective Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\dim V > \dim W$. Then no linear map from $V$ to $W$ is injective.
#review to make this intuitive.
then we can show that no map from finite-dim vec space to a bigger vec space can be surjective
wait no this one doesnt make sense. cus two elements can map to one, right? #question noo! it doesnt, cus then u would need to have a single function output multiple things to make up for it which functions can't do.
title: A map to a larger dimensional space is not surjective Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\dim V < \dim W$. Then no linear map from $V$ to $W$ is surjective.
these have imporant conseqneces! in linear equation theory.
idea: express questions about systems of linear equations in terms of linear maps
ie. use linear transformation to represent querys about linear equations
3.25.. what the hell?? #review #question srry i don't have enough brain space to interpret this right now.
title: homogenous system of linear equations A homogenous system of linear equations with more variables than equations has nonzero solutions.
oh, here, we get to define the concept of free variables
title: Inhomogenous system of linear equations An inhomogenous system of linear equations with more equations than variables has no solution for some choice of the constant terms.
these can be proved using gaussian elim!
this one needs a reflection! need to watch another vid on the concept of null space and range ect. then take notes on that as well. seek 3b1b for intutive understanding here.