Editing Row and column spaces (section)

===Dimension===
{{main|Rank (linear algebra)}}
The [[dimension (linear algebra)|dimension]] of the column space is called the '''[[rank (linear algebra)|rank]]''' of the matrix.  The rank is equal to the number of pivots in the [[reduced row echelon form]], and is the maximum number of linearly independent columns that can be chosen from the matrix.  For example, the 4&nbsp;×&nbsp;4 matrix in the example above has rank three.

Because the column space is the [[image (mathematics)|image]] of the corresponding [[matrix transformation]], the rank of a matrix is the same as the dimension of the image.  For example, the transformation <math>\R^4 \to \R^4</math> described by the matrix above maps all of <math>\R^4</math> to some three-dimensional [[Euclidean subspace|subspace]].

The '''nullity''' of a matrix is the dimension of the [[kernel (matrix)|null space]], and is equal to the number of columns in the reduced row echelon form that do not have pivots.<ref>Columns without pivots represent free variables in the associated homogeneous [[system of linear equations]].</ref>  The rank and nullity of a matrix {{mvar|A}} with {{mvar|n}} columns are related by the equation:
:<math>\operatorname{rank}(A) + \operatorname{nullity}(A) = n.\,</math>
This is known as the [[rank–nullity theorem]].