Dimension of a Subspace

Every subspace of a finite-dimensional vector space is finite-dimension (see here) and so has a dimension. The next result gives the expected inequality about the dimension of the subspace.

Dimension of a subspace

If $V$ is finite-dimensional and $U$ is a subspace of $V$, then $\dim U\le \dim V$.

Proof. Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Think of a basis of $U$ as a linearly independent list in $V$, and think of a basis of $V$ as a spanning list in $V$. Now, we can use the fact that

$$\text{length of linearly independent list}\le \text{length of spanning list}$$

to conclude that $\dim U\le \dim V$.