Dimension and rank

Objective. Use dimension, rank and nullity consistently.

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Learn: the key idea

The dimension of a vector space is the number of vectors in any basis (a well-defined invariant), while for a matrix A, the rank is the dimension of its column space (equal to the number of pivot columns) and the nullity is the dimension of its null space, linked by the rank–nullity theorem rank(A)+nullity(A)=n, where n is the number of columns. This theorem shows how pivot and free variables partition the columns of a matrix after row reduction. For a 3×4 matrix with rank 2, nullity must be 4−2=2, meaning the solution set of Ax=0 has two free parameters. Consistently tracking rank, nullity and the total column count avoids contradictions when describing solution spaces.

Worked example

A 3×5 matrix A has rank 3. Use rank and dimension facts to find the nullity of A.

1 $Recall the rank-nullity theorem: rank(A) + nullity(A) = n, where n is the number of columns.$
2 $Here n=5 (five columns) and rank(A)=3.$
3 $Rearrange: nullity(A) = n - rank(A) = 5 - 3.$
4 $Compute: nullity(A) = 2.$

Answer: 2

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Common mistake. Students compute rank from row echelon form correctly but then assume nullity equals the number of rows minus rank, rather than the number of columns minus rank; nullity is governed by the column count, since it measures dimension of the domain's null space.

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