Dimension

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

$$\text{Theorem One:}$$ If any vector space V has a basis $\mathcal{B}$ with n elements, then any set in V containing more than n vectors must be linearly dependent.

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

The dimension of V is the number of vectors in a basis for V. The dimension of {0} is 0.

Subspaces

Let H be a subspace of V. Any linearly independent set in H can be expanded to a basis for H. $$dim(H) \leq dim(V)$$

Basis Theorem

This theorem lets us make bases much more easily.

For an n-dimensional evector space V, if any set of n elements in V spans V or is linearly independent, then it's a basis.

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.