## Vector spaces and linear transformations

• A vector space over a field $𝔽$ is a set $V$ with functions  $\begin{array}{ccc}V×V& \to & V\\ \left({v}_{1},{v}_{2}\right)& ⟼& {v}_{1}+{v}_{2}\end{array}$     and     $\begin{array}{ccc}𝔽×V& \to & V\\ \left(c,v\right)& ⟼& cv\end{array}$,
addition and scalar multiplication, such that
(a)   If ${v}_{1},{v}_{2},{v}_{3}\in V$ then $\left({v}_{1}+{v}_{2}\right)+{v}_{3}=\left({v}_{1}+\left({v}_{2}+{v}_{3}\right)$,
(b)   If ${v}_{1},{v}_{2}\in V$ then ${v}_{1}+{v}_{2}={v}_{2}+{v}_{1}$,
(c)   There exists a zero, $0\in V$, such that if $v\in V$ then $0+v=v+0=v$,
(d)   If $v\in V$ then there exists an additive inverse to $v$, $-v\in V$, such that $\left(-v\right)+v=v+\left(-v\right)=0$,
(e)   If ${c}_{1},{c}_{2}\in 𝔽$ and $v\in V$ then ${c}_{1}\left({c}_{2}v\right)=\left({c}_{1}{c}_{2}\right)v$,
(f)   If $v\in V$ then $1v=v$,
(g)   If $c\in 𝔽$ and ${v}_{1},{v}_{2}\in V$ then $c\left({v}_{1}+{v}_{2}\right)=c{v}_{1}+c{v}_{2}$,
(h)   If ${c}_{1},{c}_{2}\in 𝔽$ and $v\in V$ then $\left({c}_{1}+{c}_{2}\right)v={c}_{1}v+{c}_{2}v$,

Properties (a), (b), (c) and (d) imply that a vector space $V$ is an abelian group with an action of the field $𝔽$. Hence, a vector space is just a module over a field.

Important examples of vector spaces are:

(a)   ${ℝ}^{k}$ and ${ℂ}^{k}$, for $k\in {ℤ}_{>0}$,
(b)   ${𝔽}^{k}$, where $𝔽$ is a field and $k\in {ℤ}_{>0}$.

HW: Show that is $V$ is a vector space over $𝔽$ and $v\in V$ then $0v=0$. (Notice that the $0$ on the left side of this equation is in $𝔽$ and the 0 on the right hand side is an element of $V$.)
HW: Show that if $V$ is a vector space over $𝔽$ and $c\in 𝔽$ 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 $𝔽$ be a field and let $V$ and $W$ be vector spaces over $𝔽$.

• 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 $f\left({v}_{1}+{v}_{2}\right)=f\left({v}_{1}\right)+f\left({v}_{2}\right)$,
(b)   If $c\in 𝔽$ 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 $f\left(-v\right)=-f\left(v\right)$.

• A subspace $W$ of a vector space $V$ over a field $𝔽$ 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 $c\in 𝔽$ 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 $c\in 𝔽$.

• 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 $𝔽$ 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 $𝔽$ and let $W$ be a subgroup of $V$.

• A coset of $W$ in $V$ is a set $v+W=\left\{v+w\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}w\in W\right\}$, 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 $𝔽$ 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 $↔$ Subspaces

Let $V$ be a vector space over $𝔽$ 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 $𝔽$.

Let $W$ be a subgroup of a vector space $V$ over a field $𝔽$. Then $W$ is a subspace of $V$ if and only if $V/W$ with operations given by $(v1+W) + (v2+W) = (v1+v2) +W and c (v+W) = c v+W$ is a vector space over $𝔽$.

• The quotient space $V/W$ is the vector space of cosets of a subspace $W$ of a vector space $V$ over a field $𝔽$ with operations given by $\left({v}_{1}+W\right)+\left({v}_{2}+W\right)=\left({v}_{1}+{v}_{2}\right)+W$ and $c\left(v+W\right)=cv+W$.

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 $kerf={v∈V | f(v)=0W},$ 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 $imf={f(v) | v∈V }.$

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 $f^: V/kerf ⟶ W v+N ⟼ f(v)$ Then $\stackrel{^}{f}$ is a well defined injective linear transformation.
(b)   Let $f:V\to W$ be a linear transformation and define $f′: V ⟶ imf v ⟼ f(v)$ Then $f\prime$ is a well defined surjective linear transformation.
(c)   If $f:V\to W$ is a linear transformation then $V/kerf≃imf,$ where the isomorphism is a vector space isomorphism.

#### Direct sums

Suppose $V$ and $W$ are vector spaces over a field $𝔽$. The idea is to make $V×W$ into a vector space.

• The direct sum $V\oplus W$ of two vector spaces $V$ and $W$ over a field $𝔽$ is the set $V×W$ with operations given by $(v1,w1) + (v2,w2) = (v1+v2, w1+w2) and c(v,w)= (cv,cw) ,$ for $v,{v}_{1},{v}_{2}\in V$, $w,{w}_{1},{w}_{2}\in w$ and $c\in 𝔽$. The operations in $V\oplus W$ are componentwise.
• More generally, given vector spaces ${V}_{1},{V}_{2},\dots ,{V}_{n}$ over $𝔽$ the direct sum ${V}_{1}\oplus {V}_{2}\oplus \cdots \oplus {V}_{n}$ is the set ${V}_{1}×{V}_{2}×\cdots ×{V}_{n}$ with the operations given by $(v1,…, vi,…, vn) + (w1,…, wi,…, wn) = (v1+w1 ,…, vi+wi,…, vn+wn)$ $and c(v1,…, vi,…, vn) = (cv1,…, cvi,…, cvn) ,$ where ${v}_{i},{w}_{i}\in {V}_{i}$, $c\in 𝔽$, 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 ${V}_{1}\oplus {V}_{2}\oplus \cdots \oplus {V}_{n}$ are vector spaces over $𝔽$ with zeros given by $\left({0}_{V},{0}_{W}$ and $\left({0}_{{V}_{1}},\dots ,{0}_{{V}_{n}}$, 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 1993-1994.

[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, McGraw-Hill, 1987. MR0924157.