Bases of vector spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 14 August 2011

Bases and dimension

Let $𝔽$ be a field and let $V$ be a vector space over $𝔽$. Let $\{v_1,v_2,\dots ,v_k\}$ be a subset of $V$.

• The span of the set $\{v_1,v_2,\dots ,v_k\}$ is the subspace of $V$, $$\mathrm{span}\{v_1,v_2,\dots ,v_k\}=\{c_1{v}_1+c_2{v}_2+\cdots +c_k{v}_k|c_1,c_2,\dots ,c_k\in 𝔽\}.$$
• A linear combination of $v_1,v_2,\dots ,v_k$ is an element of $\mathrm{span}\{v_1,v_2,\dots ,v_k\}$.
• The set $\{v_1,v_2,\dots ,v_k\}$ is linearly independent if it satisfies: $$\text{if}{c}_1,c_2,\dots ,c_k\in 𝔽\text{and}{c}_1{v}_1+c_2{v}_2+\cdots +c_k{v}_k=0\text{then}{c}_1=0,c_2=0,\dots ,c_k=0.$$
• 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 $\{S\subseteq V|\mathrm{span}\left(S\right)=V\}$, ordered by inclusion.

(c)   $B$ is a maximal element of $\{L\subseteq V|L\text{is linearly independent}\}$, 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 $$v=\sum _{s\in S}{r}_{s}s,$$
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 $$\sum _{s\in S}{r}_{s}s=0\text{then}{r}_s=0,\text{for all}s\in S.$$
• 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 $\{v_1,v_2,\dots ,v_n\}$ 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 $C\times B$ matrix with entries in $R$ is a collection $F=\left(F_{cb}\right)$ of elements $F_{cb}\in R$ indexed by the elements of $C\times B$ and such that for each $b\in B$ all but a finite number of the $F_{cb}$ are equal to 0. $$M_{C\times B}\left(R\right)=\{C\times B\text{matrices with entries in}R\}.$$
• The sum of two matrices $F_1,F_2\in M_{C\times B}\left(R\right)$ is the matrix $F_1+F_2$ given by $${(F_1+F_2)}_{cb}={\left(F_1\right)}_{cb}+{\left(F_2\right)}_{cb},\text{for}b\in B\text{and}c\in C.$$
• The product of matrices $F_1\in M_{D\times C}\left(R\right)$ and $F_2\in M_{C\times B}\left(R\right)$ is the matrix $F_1{F}_2\in M_{D\times B}\left(R\right)$ given by $${\left(F_1{F}_2\right)}_{db}=\sum _{c\in C}{\left(F_1\right)}_{dc}{\left(F_2\right)}_{cb},\text{for}b\in B\text{and}d\in D.$$

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 $f_{CB}\in M_{C\times B}\left(R^{\mathrm{op}}\right)$ given by $${\left(f_{CB}\right)}_{cb}=f_{cb},\text{where}{f}_{cb}\in R^{\mathrm{op}}\text{are given by}f\left(b\right)=\sum _{c\in C}{f}_{cb}c,$$ 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 $P_{CB}\in M_{C\times B}\left(R^{\mathrm{op}}\right)$ given by $${\left(P_{CB}\right)}_{cb}=P_{cb},\text{where}{P}_{cb}\in R^{\mathrm{op}}\text{are given by}b=\sum _{c\in C}{P}_{cb}c,$$ for $b\in B$.

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

(a)   The function $$\begin{array}{ccc}{\mathrm{Hom}}_R(M,N)& \to & M_{C\times B}\left(R^{\mathrm{op}}\right)\\ f& ⟼& f_{CB}\end{array}$$ is an isomorphism of abelian groups.

(b)   $$\begin{array}{ccc}{\mathrm{End}}_R\left(M\right)& \to & M_{B\times B}\left(R^{\mathrm{op}}\right)\\ f& ⟼& f_{CB}\end{array}$$ 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 $$f_{C_2{B}_2}=Q{f}_{C_1{B}_1}P.$$

(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 $$f_{B_2}=P^{-1}{f}_{B_1}P.$$

Proofs

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

(a)   The function $$\begin{array}{ccc}{\mathrm{Hom}}_R(M,N)& \to & M_{C\times B}\left(R^{\mathrm{op}}\right)\\ f& ⟼& f_{CB}\end{array}$$ is an isomorphism of abelian groups.

(b)   $$\begin{array}{ccc}{\mathrm{End}}_R\left(M\right)& \to & M_{B\times B}\left(R^{\mathrm{op}}\right)\\ f& ⟼& f_{CB}\end{array}$$ 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 $f\in {\mathrm{Hom}}_R(M,N)$ and $g\in {\mathrm{Hom}}_R(N,P)$ then ${\left(fg\right)}_{DB}=f_{DC}{g}_{CB}$. $$f_1\left(f_2\right(b\left)\right)=\sum _{c\in C}{\left(f_2\right)}_{cb}{f}_1\left(c\right)=\sum _{c\in C}\sum _{d\in D}{\left(f_2\right)}_{cb}{\left(f_1\right)}_{dc}d=\sum _{d\in D}\left(\sum _{c\in C}{\left(f_2\right)}_{cb}{\left(f_1\right)}_{dc}\right)d.$$ So $${\left(f_1{f}_2\right)}_{db}={\left(\sum _{c\in C}{\left(f_2\right)}_{cb}{\left(f_1\right)}_{dc}\right)}^{\mathrm{op}}={\left(f_1\right)}_{dc}{\left(f_2\right)}_{cb}={\left(\left(f_1\right)\left(f_2\right)\right)}_{db}.$$

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.

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