## Bases and dimension

Let $𝔽$ be a field and let $V$ be a vector space over $𝔽$. Let $\left\{{v}_{1},{v}_{2},\dots ,{v}_{k}\right\}$ be a subset of $V$.

• The span of the set $\left\{{v}_{1},{v}_{2},\dots ,{v}_{k}\right\}$ is the subspace of $V$, $span{ v1, v2, …, vk } = {c1v1 + c2v2 +⋯+ ckvk | c1, c2, …, ck ∈𝔽}.$
• A linear combination of ${v}_{1},{v}_{2},\dots ,{v}_{k}$ is an element of $\mathrm{span}\left\{{v}_{1},{v}_{2},\dots ,{v}_{k}\right\}$.
• The set $\left\{{v}_{1},{v}_{2},\dots ,{v}_{k}\right\}$ is linearly independent if it satisfies: $if c1, c2, …, ck ∈𝔽 and c1v1 + c2v2 +⋯+ ckvk =0 then c1=0, c2=0, …, ck =0.$
• A basis of $V$ is a subset $B\subseteq V$ such that
(a)   $\mathrm{span}\left(B\right)=V$,
(b)   $B$ is linearly independent.
• The dimension of $V$ is the cardinality (number of elements) of a basis of $V$.

Let $B$ be a subset of $V$. The following are equivalent

(a)   $B$ is a basis of $V$.
(b)   $B$ is a minimal element of $\left\{S\subseteq V\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mathrm{span}\left(S\right)=V\right\}$, ordered by inclusion.
(c)   $B$ is a maximal element of $\left\{L\subseteq V\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}L\phantom{\rule{.5em}{0ex}}\text{is linearly independent}\right\}$, ordered by inclusion.

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

(a)   Then $V$ has a basis.
(b)   Any two bases of $V$ have the same number of elements.

Let $R$ be a ring and let $M$ be an $R$-module. Let $S$ be a subset of $M$.

• The submodule generated by $S$ is the smallest submodule of $M$ containing $S$, i.e. the submodule generated by $S$ is the submodule $\mathrm{span}\left(S\right)$ of $M$ such that
(a)   $S\subseteq \mathrm{span}\left(S\right)$,
(b)   If $N$ is a submodule of $M$ such that $S\subseteq N$ then $\mathrm{span}\left(S\right)\subseteq N$.
• The $R$-module $M$ is finitely generated if there is a finite subset $S$ of $M$ such that $\mathrm{span}\left(S\right)=M$.
• A linear combination of elements of $S$ is an element $v\in M$ of the form $v= ∑s∈S rss,$
• where ${r}_{s}\in R$ and ${r}_{s}=0$ for all but a finite number of $s\in S$.
• The set $S$ is linearly independent if it satisfies the condition: if ${r}_{s}\in R$ and ${r}_{s}=0$ for all but a finite number of $s\in S$, and $∑s∈S rss=0 then rs=0, for all s∈S.$
• A basis of $M$ is a subset $B\subseteq M$ such that
(a)   $\mathrm{span}\left(B\right)=M$,
(b)   $B$ is linearly independent.
• A free module is an $R$-module $M$ that has a basis.

HW: Let $M$ be an $R$-module and let $S\subseteq M$ be a subset of $M$. Show that $\mathrm{span}\left(S\right)$ exists and is unique by showing that $\mathrm{span}\left(S\right)$ is the intersection of all the submodules that contain $S$.

HW: Show that $\mathrm{span}\left(S\right)$ is the set of linear combinations of $S$.

HW: Let $M$ be an $R$-module. Show that a subset $S\subseteq M$ is linearly independent if and only if $S$ satisfies the following property:

HW: If ${r}_{m}\in R$ such that ${r}_{m}=0$ for all but a finite number of $m\in M$ and $\sum _{m\in S}{r}_{m}m=0$ then ${r}_{m}$ for all $m\in M$.

HW: Let $V$ be a vector space over a field $𝔽$. Show that if $v\in V$ and $\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}$ is a basis of $V$ then there exist unique ${c}_{i}\in 𝔽$ such that $v={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\cdots +{c}_{n}{v}_{n}$.

HW: Let $R$ be a commutative ring. Give an example of a finitely generated $R$-module that does not have a basis.

HW: Give an example of a ring $R$ and a finitely generated module over $R$ that has two different bases with different numbers of elements.

HW: Give an example of a finitely generated module that is not free.

HW: Give an example of a free module that is not finitely generated.

HW: Show that every module over a field is free.

HW: Show that $R$ is a division ring if and only if every $R$-module is free.

(a)   Let $R$ be a ring and let $M$ be a free $R$-module with an infinite basis. Any two bases of $M$ have the same number of elements.
(b)   Let $R$ be a commutative ring and let $M$ be a free $R$-module. Any two bases of $M$ have the same number of elements.

Let $B,C$ and $D$ be sets. Let $R$ be a ring.

• A $C×B$ matrix with entries in $R$ is a collection $F=\left({F}_{cb}\right)$ of elements ${F}_{cb}\in R$ indexed by the elements of $C×B$ and such that for each $b\in B$ all but a finite number of the ${F}_{cb}$ are equal to 0. $MC×B( R)= {C×B matrices with entries inR} .$
• The sum of two matrices ${F}_{1},{F}_{2}\in {M}_{C×B}\left(R\right)$ is the matrix ${F}_{1}+{F}_{2}$ given by $( F1+F2) cb = ( F1) cb + (F2) cb, for b∈B and c∈C.$
• The product of matrices ${F}_{1}\in {M}_{D×C}\left(R\right)$ and ${F}_{2}\in {M}_{C×B}\left(R\right)$ is the matrix ${F}_{1}{F}_{2}\in {M}_{D×B}\left(R\right)$ given by $( F1F2) db = ∑c∈C ( F1) dc (F2) cb, for b∈B and d∈D.$

Let $M$ and $N$ be free $R$-modules with bases $B$ and $C$, respectively. Let $f:M\to N$ be a homomorphism.

• The matrix of $f:M\to N$ with respect to the bases $B$ and $C$ is the matrix ${f}_{CB}\in {M}_{C×B}\left({R}^{\mathrm{op}}\right)$ given by $( fCB ) cb = fcb, where fcb∈ Rop are given by f(b) = ∑c∈C fcbc,$ for $b\in B$.

Let $M$ be a free $R$-module and let $B$ and $C$ be bases of $M$.

• The change of basis matrix from $B$ to $C$ is the matrix ${P}_{CB}\in {M}_{C×B}\left({R}^{\mathrm{op}}\right)$ given by $( PCB ) cb = Pcb, where Pcb∈ Rop are given by b = ∑c∈C Pcbc,$ for $b\in B$.

Let $M$ and $N$ be free $R$-modules with bases $B$ and $C$, respectively.

(a)   The function $HomR(M,N) → MC×B( Rop) f ⟼ fCB$ is an isomorphism of abelian groups.
(b)   $EndR(M) → MB×B( Rop) f ⟼ fCB$ is a ring isomorphism.

HW: Discuss the difficulties in trying to make the map in Proposition ??? (a) into an $R$-module homomorphism, or into an ${R}^{\mathrm{op}}$-module homomorphism.

HW: Let $M$ be a free $R$-module. Let $B$ and $C$ be bases of $M$. Let $P$ be the change of basis matrix from $B$ to $C$ and let $Q$ be the change of basis matrix from $C$ to $B$. Show that $Q={P}^{-1}$.

(a)   Let $f:M\to N$ be an $R$-module homomorphism. Let ${B}_{1}$ and ${B}_{2}$ be bases of $M$ and let $P$ be the change of basis matrix from ${B}_{2}$ to ${B}_{1}$. Let ${C}_{1}$ and ${C}_{2}$ be bases of $N$ and let $Q$ be the change of basis matrix from ${C}_{1}$ to ${C}_{2}$. Then $fC2B2 = Q fC1B1 P.$
(b)   Let $f:M\to M$ be an $R$-module homomorphism. Let ${B}_{1}$ and ${B}_{2}$ be bases of $M$ and let $P$ be the change of basis matrix from ${B}_{2}$ to ${B}_{1}$. Then $fB2 = P-1 fB1 P.$

## Proofs

Let $M$ and $N$ be free $R$-modules with bases $B$ and $C$, respectively.

(a)   The function $HomR(M,N) → MC×B( Rop) f ⟼ fCB$ is an isomorphism of abelian groups.
(b)   $EndR(M) → MB×B( Rop) f ⟼ fCB$ is a ring isomorphism.

Proof.

In fact we shall show that if $M,N$ and $P$ are free modules with bases $B,C$ and $D$, respectively, and if $f\in {\mathrm{Hom}}_{R}\left(M,N\right)$ and $g\in {\mathrm{Hom}}_{R}\left(N,P\right)$ then ${\left(fg\right)}_{DB}={f}_{DC}{g}_{CB}$. $f1( f2(b)) = ∑c∈C (f2) cb f1(c) = ∑c∈C ∑d∈D (f2) cb (f1) dc d = ∑d∈D ( ∑c∈C (f2) cb (f1) dc ) d.$ So $(f1f2) db = ( ∑c∈C (f2) cb (f1) dc ) op = (f1) dc (f2) cb = ( (f1) (f2) ) db .$

## 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.