Theorem

If a vector space $V$ has a basis $\mathcal{B} = \{ \vec{b}_1, \dots, \vec{b}_n \}$, then any set in $V$ containing more than $n$ vectors must be linearly dependent.

Theorem

If a vector space $V$ has a basis of $n$ vectors, then every basis of $V$ must consist of exactly $n$ vectors.

Theorem

Let $H$ be a subspace of a finite-dimensional vector space $V$. Any linearly independent set in $H$ can be expanded, if necessary, to a basis for $H$. Also, $H$ is finite-dimensional and

$$\dim H \leq \dim V$$

The Basis Theorem

Let $V$ be a $p$-dimensional vector space, $p \geq 1$. Any linearly independent set of exactly $p$ elements in $V$ is automatically a basis for $V$. Any set of exactly $p$ elements that spans $V$ is automatically a basis for $V$.

Theorem

The dimension of Nul $A$ is the number of free variables in the equation $A\vec{x} = \vec{0}$, and the dimension of Col $A$ is the number of pivot columns in $A$.