In this entry we look at various formulae involving the dimension of a vector space.

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.

All of these formulae potentially involve infinite cardinals, so the reader should have a basic knowledge of cardinal arithmetic in order to understand them in full generality.

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 $\phi$ :$$ \dim(V)=\dim(\Im\phi)+\dim(\Ker\phi).$$

In particular, if $U$ is a subspace of $V$ then$$ \dim(V)=\dim(V/U)+\dim(U).$$

The rank-nullity theorem can also be stated in terms of short exact sequences: if$$ 0\longrightarrow U\longrightarrow V\longrightarrow W\longrightarrow 0$$ is a short exact sequence of vector spaces over $K$ , then$$ \dim(V) = \dim(U) + \dim(W).$$ This can be generalized to infinite exact sequences: if$$ \cdots\longrightarrow V_{n+1}\longrightarrow V_n\longrightarrow V_{n-1}\longrightarrow \cdots$$ 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:

The effect of the above formula is somewhat different depending on whether $K$ is finite or infinite. If $K$ is finite, then it reduces to

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

More generally, we can consider the space $V^S$ , which is really just the direct product of copies of $V$ indexed by $S$ . We get

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 $V^*$ :

Space of linear mappings

The set $\Hom_K(V,W)$ 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 $f,g\in\Hom_K(V,W)$ , all $x\in V$ , and all $\lambda\in K$ . The dual space $V^*=\Hom_K(V,K)$ 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 $\Hom_K(V,W)$ and $W^B$ , so that from an earlier section we get

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

Tensor products

The dimension of the tensor product 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[3].

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

References

1

Reinhold Baer, Linear Algebra and Projective Geometry, 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.