Bases

Overview

The dimension of a finite-dimensional vector space $V$ is the length of any basis of the vector space. The dimension of $V$ is denoted $dim V$ .

If $U$ is a subspace of $V$ , then $dim U \leq dim V$ .

Let $V$ be a finite-dimensional vector space and $T \in L (V, W)$ . Then $Im T$ is finite-dimensional and

dim V = nullity T + rank T .

This result is also known as the fundamental theorem of linear maps.

Let $V$ and $W$ be finite-dimensional vector spaces. If $dim V > dim W$ , then no linear map from $V$ to $W$ is injective.

Let $V$ and $W$ be finite-dimensional vector spaces. If $dim V < dim W$ , then no linear map from $V$ to $W$ is surjective.

Let $V$ be a finite-dimensional vector space. If $U_{1}$ and $U_{2}$ are subspaces of $V$ , then

dim (U_{1} + U_{2}) = dim U_{1} + dim U_{2} - dim (U_{1} \cap U_{2}) .

Suppose $U_{1}, \dots, U_{m}$ are finite-dimensional subspaces of some vector space. Then

dim (U_{1} + \dots + U_{m}) \leq dim U_{1} + \dots + dim (U_{m}) .

The above inequality forms an equality if and only if $U_{1} + \dots + U_{m}$ is a direct sum.

Suppose $V_{1}, \dots, V_{m}$ are finite-dimensional vector spaces. Then external direct product $\prod V_{i}$ is finite-dimensional and

dim (V_{1} \times \dots \times V_{m}) = dim V_{1} + \dots + dim V_{m} .

Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$ . Then

dim V / U = dim V - dim U .