# dimension formulae for vector spaces

Throughout this entry, $K$ will be a field, and $V$ and $W$ will be vector spaces over $K$. The dimension of a vector space $U$ over $K$ will be denoted by $\dim(U)$, or by $\dim_{K}(U)$ if the ground field needs to be emphasized.

## Subspaces

If $S$ and $T$ are subspaces  of $V$, then

 $\dim(S)+\dim(T)=\dim(S\cap T)+\dim(S+T).$

## Rank-nullity theorem

The rank-nullity theorem  states that if $\phi\colon V\to W$ is a linear mapping, then the dimension of $V$ is the sum of the dimensions of the image and kernel of $\phi$:

 $\dim(V)=\dim(\operatorname{Im}\phi)+\dim(\operatorname{Ker}\phi).$

In particular, if $U$ is a subspace of $V$ then

 $\dim(V)=\dim(V/U)+\dim(U).$
 $0\longrightarrow U\longrightarrow V\longrightarrow W\longrightarrow 0$

is a short exact sequence of vector spaces over $K$, then

 $\dim(V)=\dim(U)+\dim(W).$
 $\cdots\longrightarrow V_{n+1}\longrightarrow V_{n}\longrightarrow V_{n-1}\longrightarrow\cdots$

is an exact sequence of vector spaces over $K$, then

 $\sum_{n\rm{~{}even}}\!\!\dim(V_{n})\,=\sum_{n\rm{~{}odd}}\!\dim(V_{n}).$

(This is indeed a generalization  , because any finite exact sequence of vectors spaces terminating with $0$ at both ends can be considered as an infinite exact sequence in which all remaining terms are $0$.)

## Direct sums

If $(V_{i})_{i\in I}$ is a family of vector spaces over $K$, then

 $\dim\!\left(\bigoplus_{i\in I}V_{i}\right)=\sum_{i\in I}\dim(V_{i}).$

## Cardinality of a vector space

The cardinality of a vector space is determined by its dimension and the cardinality of the ground field:

 $|V|=\begin{cases}|K|^{\dim(V)},&\hbox{if }\dim(V)\hbox{ is finite};\\ \max\{|K|,\dim(V)\},&\hbox{if }\dim(V)\hbox{ is infinite}.\cr\end{cases}$

The effect of the above formula   is somewhat different depending on whether $K$ is finite (http://planetmath.org/FiniteField) or infinite. If $K$ is finite, then it reduces to

 $|V|=\begin{cases}|K|^{\dim(V)},&\hbox{if }\dim(V)\hbox{ is finite};\\ \dim(V),&\hbox{if }\dim(V)\hbox{ is infinite}.\end{cases}$

If $K$ is infinite, then it can be expressed as

 $|V|=\begin{cases}1,&\hbox{if }\dim(V)=0;\\ |K|,&\hbox{if }0<\dim(V)\leq|K|;\\ \dim(V),&\hbox{if }\dim(V)\geq|K|.\end{cases}$

## Change of ground field

If $F$ is a subfield of $K$, then $V$ can be considered as a vector space over $F$. The dimensions of $V$ over $K$ and $F$ are related by the formula

 $\dim_{F}(V)=[K:F]\cdot\dim_{K}(V).$

In this formula, $[K:F]$ is the degree of the field extension $K/F$, that is, the dimension of $K$ considered as a vector space over $F$.

## Space of functions into a vector space

If $S$ is any set, then the set $K^{S}$ of all functions from $S$ into $K$ becomes a vector space over $K$ if we define the operations  pointwise, that is, $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x)$ for all $f,g\in K^{S}$, all $x\in S$, and all $\lambda\in K$. The dimension of this vector space is given by

 $\dim(K^{S})=\begin{cases}|S|,&\hbox{if }S\hbox{ is finite};\\ |K|^{|S|},&\hbox{if }S\hbox{ is infinite}.\end{cases}$

The case where $S$ is infinite is not straightforward to prove. Proofs can be found in books by Baer and Jacobson, among others.

## Dual space

Given any basis $B$ of $V$, the dual space    $V^{*}$ is isomorphic   to $K^{B}$ via the mapping $f\mapsto f|_{B}$. So the formula of the previous section      can be applied to give a formula for the dimension of $V^{*}$:

 $\dim(V^{*})=\begin{cases}\dim(V),&\hbox{if }\dim(V)\hbox{ is finite};\\ |K|^{\dim(V)},&\hbox{if }\dim(V)\hbox{ is infinite}.\end{cases}$

In particular, this formula implies that $V$ is isomorphic to $V^{*}$ if and only if $V$ is finite-dimensional. (Students who are familiar with the fact that an infinite-dimensional Banach space can be isomorphic to its dual are sometimes surprised to learn that an infinite-dimensional vector space cannot be isomorphic to its dual, for a Banach space is surely a vector space. But the term dual is used in different senses in these two statements, so there is no contradiction   . In the theory of Banach spaces one is usually only interested in the linear functionals    , and the resulting ‘continuous’ dual is a subspace of the full dual used in the above formula.)

## Space of linear mappings

The set $\operatorname{Hom}_{K}(V,W)$ of all linear mappings from $V$ into $W$ is itself a vector space over $K$, with the operations defined in the obvious way, namely $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x)$ for all $f,g\in\operatorname{Hom}_{K}(V,W)$, all $x\in V$, and all $\lambda\in K$. The dual space $V^{*}=\operatorname{Hom}_{K}(V,K)$ considered in the previous section is a special case of this. For any basis $B$ of $V$, the mapping $f\mapsto f|_{B}$ defines an isomorphism      between $\operatorname{Hom}_{K}(V,W)$ and $W^{B}$, so that from an earlier section we get

 $\dim(\operatorname{Hom}_{K}(V,W))=\begin{cases}0,&\hbox{if }\dim(W)=0;\\ \dim(V)\cdot\dim(W),&\hbox{if }\dim(V)\hbox{ is finite};\\ |W|^{\dim(V)},&\hbox{otherwise}.\end{cases}$

In the special case $W=V$ this can be simplified to

 $\dim(\operatorname{End}_{K}(V))=\begin{cases}\dim(V)^{2},&\hbox{if }\dim(V)% \hbox{ is finite};\\ |K|^{\dim(V)},&\hbox{otherwise}.\end{cases}$

## Tensor products

The dimension of the tensor product   (http://planetmath.org/TensorProduct) of $V$ and $W$ is given by

 $\dim(V\otimes W)=\dim(V)\cdot\dim(W).$

## Banach spaces

The dimension of a Banach space, considered as a vector space, is sometimes called the Hamel dimension, in order to distinguish it from other concepts of dimension. For an infinite-dimensional Banach space $B$ we have

 $\dim(B)=|B|.$

The tricky part of establishing this formula is to show that the dimension is always at least the cardinality of the continuum  . A short proof of this is given in a paper by Lacey.

The above formula suggests that Hamel dimension is not a very useful concept for infinite-dimensional Banach spaces, which is indeed the case. Nonetheless, it is interesting to see how Hamel dimension relates to the usual concept of dimension in Hilbert spaces  . If $H$ is a Hilbert space, and $d$ is its dimension (meaning the cardinality of an orthonormal basis), then the Hamel dimension $\dim(H)$ is given by

 $\dim(H)=\begin{cases}d,&\hbox{if }d\hbox{ is finite};\\ d^{\aleph_{0}},&\hbox{if }d\hbox{ is infinite}.\end{cases}$

## References

• 1 Reinhold Baer, , Academic Press, 1952.
• 2 Nathan Jacobson, Lectures in Abstract Algebra, Volume II: Linear Algebra, D. Van Nostrand Company Inc., 1953.
• 3 H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional Separable Banach Space is c, Amer. Math. Mon. 80 (1973), 298.
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