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# 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

MathJax TeX Test Page $$\text{Theorem One:}$$ 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

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

# Subspaces

MathJax TeX Test Page 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.

MathJax TeX Test Page 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.