# Usual Definition

When we think of dimension, we think of the vector <a,b,c> having three dimensions. However, the vector space formed by <a,b,c> only has one dimension. So, what's the real definition?

## A Few Theorems First

Before we get to the real definition, let's list two theorems

This lets us say: $$\text{Theorem Two:}$$ If a vector space has a basis of n vectors, then every basis of V contains exactly n vectors.

# Real Definition

# Subspaces

# Basis Theorem

This theorem lets us make bases much more easily.

Now, we don't have to show both span and linear independence, as we had to before.

# Dimensions of Null Space and Column Space

The dimension of Nul(A) is the number of free variables (non-pivot columns).

The dimension of Col(A) is the number of pivot columns.

This will become more important in the next section.