HW: Show that is $V$ is a vector space over $\mathbb{F}$ and
$v\in V$ then $0v=0$.
(Notice that the $0$ on the left side of this equation is in $\mathbb{F}$
and the 0 on the right hand side is an element of $V$.)
HW: Show that if $V$ is a vector space over $\mathbb{F}$ and
$c\in \mathbb{F}$ and $v\in V$
then $cv=0$ if and only if either $c=0$ or $v=0$.
Linear transformations are for comparing vector spaces.
Let $\mathbb{F}$ be a field and let
$V$ and $W$ be vector spaces over $\mathbb{F}$.

A linear transformation from $V$ to $W$
is a function $f:V\to W$ such that
 (a)
If ${v}_{1},{v}_{2}\in V$ then $f({v}_{1}+{v}_{2})=f\left({v}_{1}\right)+f\left({v}_{2}\right)$,
 (b)
If $c\in \mathbb{F}$ and $v\in V$
then $f\left(cv\right)=cf\left(v\right)$.

A vector space isomorphism is a bijective linear transformation.

Two vector spaces $V$ and $W$ are isomorphic,
$V\simeq W$, is there exists a vector space isomorphism
$f:V\to W$ between them.
Two vector spaces are isomorphic if the elements of the vector spaces and the operations and the actions match up exactly. Think of two vector spaces that are isomorphic as being "the same".
HW: Let $f:V\to W$ be a linear transformation.
Show that $f\left(0\right)=0$. (Notice that the
0 on the left hand side of this equation is in $V$ and the 0 on the right hand side
is an element of $W$.
HW: Let $f:V\to W$ be a linear transformation.
Show that if $v\in V$ then
$f(v)=f\left(v\right)$.
 A subspace $W$ of a vector space $V$
over a field $\mathbb{F}$ is a subset $W\subseteq V$
such that
 (a)
If If ${w}_{1},{w}_{2}\in W$ then
${w}_{1}+{w}_{2}\in W$,
 (b)
$0\in W$,
 (c)
If $w\in W$ then
$w\in W$,
 (d)
If $w\in W$ and $c\in \mathbb{F}$
then $cw\in W$.
 The zero space, $\left(0\right)$, is the set containing
only $0$ with operations $0+0=0$ and $c\cdot 0=0$, for
$c\in \mathbb{F}$.
 Let $V$ be a vector space and let $S$ be a subset of
$V$. The span of $S$,
$\mathrm{span}\left(S\right)$, or the subspace generated by
$S$, is the subspace of $V$ such that
 (a)
$S\subseteq \mathrm{span}\left(S\right)$,
 (b)
If $W$ is a subspace of $V$ and $S\subseteq W$ then $\mathrm{span}\left(S\right)\subseteq W$.
The subspace $\mathrm{span}\left(S\right)$ is the smallest subspace of
$V$ containing $S$. Think of $\mathrm{span}\left(S\right)$ as gotten by adding to $S$ exactly those elements of $V$
that are needed to make a subspace.
Cosets
 A subgroup of a vector space $V$ over a field
$\mathbb{F}$ is a subset $W\subseteq V$
such that
 (a)
If If ${w}_{1},{w}_{2}\in W$ then
${w}_{1}+{w}_{2}\in W$,
 (b)
$0\in W$,
 (c)
If $w\in W$ then
$w\in W$,
Let $V$ be a vector space over $\mathbb{F}$
and let $W$ be a subgroup of $V$.

A coset of $W$ in $V$
is a set $v+W=\{v+w\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}w\in W\}$, where
$v\in V$.
 $V/W$ (pronounced "$V$ mod
$W$") is the set of cosets of $W$ in $V$.
Let $V$ be a vector space over a field $\mathbb{F}$ and let
$W$ be a subgroup of $V$. Then the cosets of $W$
in $V$ partition $V$.
Notice that the proofs of Proposition (vsptn) and Proposition
(gpptn) are essentially the same.
HW: Write a very shrot proof of Proposition (vsptn) by using (gpptn).
Quotient spaces $\leftrightarrow $ Subspaces
Let $V$ be a vector space over $\mathbb{F}$ and
let $W$ be a subgroup of $V$. We can try to make the set
$V/W$ of cosets of $W$ in $V$
into a vector space by defining an addition operation and an action of $\mathbb{F}$.
Let $W$ be a subgroup of a vector space $V$
over a field $\mathbb{F}$. Then $W$ is a subspace of $V$
if and only if $V/W$ with operations given by
$$({v}_{1}+W)+({v}_{2}+W)=({v}_{1}+{v}_{2})+W\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}c(v+W)=cv+W$$
is a vector space over $\mathbb{F}$.
 The quotient space $V/W$ is the vector
space of cosets of a subspace $W$ of a vector space $V$ over
a field $\mathbb{F}$ with operations given by
$({v}_{1}+W)+({v}_{2}+W)=({v}_{1}+{v}_{2})+W$
and
$c(v+W)=cv+W$.
We have made $V/W$ into a vector space when $W$
is a subspace of $V$.
kernel and image of a linear transformation
 The kernel, or null space, of a linear transformation
$f:V\to W$ is the set
$$\mathrm{ker}f=\{v\in V\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}f\left(v\right)={0}_{W}\},$$
where ${0}_{W}$ is the zero element of $W$.
 The image, or range, of a of a linear transformation
$f:V\to W$ is the set
$$\mathrm{im}f=\left\{f\right(v\left)\phantom{\rule{0.5em}{0ex}}\right\phantom{\rule{0.5em}{0ex}}v\in V\}.$$
Let $f:V\to W$ be a linear transformation.
Then
 (a)
$\mathrm{ker}f$ is a subspace of $V$.
 (b)
$\mathrm{im}f$ is a subspace of $W$.
Let $f:V\to W$ be a linear transformation.
Let ${0}_{V}$ be the zero element of $V$.
Then
 (a)
$\mathrm{ker}f=\left\{{0}_{V}\right\}$
if and only if
$f$ is injective.
 (b)
$\mathrm{im}f=W$ if and only if
$f$ is surjective.
Notice that the proof of Proposition (vsinjsur)(b) does not use the fact that
$f:V\to W$ is a linear transformation
only the fact that
$f:V\to W$ is a function.
 (a)
Let
$f:V\to W$ be a linear transformation
and let $N=\mathrm{ker}f$. Define
$$\begin{array}{cccc}\hat{f}:& V/\mathrm{ker}f& \u27f6& W\\ & v+N& \u27fc& f\left(v\right)\end{array}$$
Then $\hat{f}$ is a well defined injective linear transformation.
 (b)
Let
$f:V\to W$ be a linear transformation
and define
$$\begin{array}{cccc}f\prime :& V& \u27f6& \mathrm{im}f\\ & v& \u27fc& f\left(v\right)\end{array}$$
Then $f\prime $ is a well defined surjective linear transformation.
 (c)
If
$f:V\to W$ is a linear transformation
then
$$V/\mathrm{ker}f\simeq \mathrm{im}f,$$
where the isomorphism is a vector space isomorphism.
Direct sums
Suppose $V$ and $W$ are vector spaces over a field
$\mathbb{F}$. The idea is to make $V\times W$
into a vector space.
 The direct sum $V\oplus W$ of two vector
spaces $V$ and $W$ over a field $\mathbb{F}$
is the set $V\times W$ with operations given by
$$({v}_{1},{w}_{1})+({v}_{2},{w}_{2})=({v}_{1}+{v}_{2},{w}_{1}+{w}_{2})\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}c(v,w)=(cv,cw),$$
for $v,{v}_{1},{v}_{2}\in V$,
$w,{w}_{1},{w}_{2}\in w$ and
$c\in \mathbb{F}$.
The operations in $V\oplus W$ are componentwise.
 More generally, given vector spaces ${V}_{1},{V}_{2},\dots ,{V}_{\mathrm{n}}$
over $\mathbb{F}$ the direct sum
${V}_{1}\oplus {V}_{2}\oplus \cdots \oplus {V}_{\mathrm{n}}$
is the set
${V}_{1}\times {V}_{2}\times \cdots \times {V}_{\mathrm{n}}$
with the operations given by
$$({v}_{1},\dots ,{v}_{i},\dots ,{v}_{n})+({w}_{1},\dots ,{w}_{i},\dots ,{w}_{n})=({v}_{1}+{w}_{1},\dots ,{v}_{i}+{w}_{i},\dots ,{v}_{n}+{w}_{n})$$
$$\text{and}\phantom{\rule{2em}{0ex}}c({v}_{1},\dots ,{v}_{i},\dots ,{v}_{n})=(c{v}_{1},\dots ,c{v}_{i},\dots ,c{v}_{n}),$$
where ${v}_{i},{w}_{i}\in {V}_{i}$,
$c\in \mathbb{F}$, and
${v}_{i}+{w}_{i}$
and $c{v}_{i}$ are given by the operations in
${V}_{i}$.
HW: Show that these are good definitions, i.e. that, as defined above,
$V\oplus W$ and
${V}_{1}\oplus {V}_{2}\oplus \cdots \oplus {V}_{\mathrm{n}}$
are vector spaces over $\mathbb{F}$ with zeros given by
$({0}_{V},{0}_{W}$
and
$({0}_{{V}_{1}},\dots ,{0}_{{V}_{n}}$, respectively.
(${0}_{{V}_{i}}$ denotes the zero element in
${V}_{i}$.)
Notes and References
These notes are written to highlight the analogy between groups and group actions,
rings and modules, and fields and vector spaces.
References
[Ram]
A. Ram,
Notes in abstract algebra,
University of Wisconsin, Madison 19931994.
[Bou]
N. Bourbaki,
Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques,
Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp.
MR0107661.
[Ru]
W. Rudin,
Real and complex analysis, Third edition, McGrawHill, 1987.
MR0924157.
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