Dimension

The dimension of a Vector Space is the number of elements in any Basis of the vector space.

Definition §

Let $V_{\mathbb{F}}$ be a vector space, with a basis for $V_{\mathbb{F}}$ being the set $S$

$$\dim (V_{\mathbb{F}})=|S|=\text{No. of elements in S}$$

If $S$ is a finite set, then $V_{\mathbb{F}}$ is finite dimensional. Else, $V_{\mathbb{F}}$ is infinite dimensional

Theorem: Finite Dimensional Vector Spaces §

Let $V_{\mathbb{F}}$ be a vector space, and suppose that $\text{dim}(V_{\mathbb{F}})$ is finite. Then:

• If a set with exactly n elements spans V , then it is a basis for V .
• If a linearly independent subset of V has exactly $n$ elements, then it is a basis for V .