AcademyVector Spaces
Academy
Nullity
Level 1 - Math I (Physics) topic page in Vector Spaces.
Principle
The nullity of a matrix is the dimension of its null space. The null space contains all input vectors that the matrix sends to the zero vector.
In a linear physics model, nullity counts independent input directions that produce no output. These may represent unconstrained modes, hidden degrees of freedom, or changes that do not affect measured quantities.
Notation
\(A\)
a matrix
\(\operatorname{Null}(A)\)
the null space of A
\(\operatorname{nullity}A\)
the dimension of the null space of A
\(A\mathbf x=\mathbf 0\)
the homogeneous system defining the null space
\(free variable\)
a variable not fixed by a pivot column
\(pivot variable\)
a variable corresponding to a pivot column
\(n\)
the number of columns of A
The number \(n\) in rank-nullity is the number of input coordinates, so it is the number of columns of \(A\), not the number of rows.
Method
- Write the homogeneous system \(A\mathbf x=\mathbf 0\).
- Row-reduce the augmented system, or row-reduce \(A\) because the right-hand side is zero.
- Identify pivot variables and free variables.
- Express the solution in terms of the free variables.
- Build a basis for the null space from those solution directions.
- Count the basis vectors; that count is the nullity.
Rules
Null space
\[\operatorname{Null}(A)=\{\mathbf x:A\mathbf x=\mathbf 0\}\]
Nullity
\[\operatorname{nullity}A=\dim\operatorname{Null}(A)\]
Rank-nullity theorem
\[\operatorname{rank}A+\operatorname{nullity}A=n\]
- Nullity equals the number of free variables in the homogeneous system.
- If a matrix has full column rank, its nullity is \(0\).
- If \(A\) has \(n\) columns and rank \(r\), then \(\operatorname{nullity}A=n-r\).
- The zero vector is always in the null space, but that does not mean the nullity is always non-zero.
Examples
Question
A matrix has 3 columns and rank 2. What is its nullity?
Answer
Use rank-nullity with
\[n=3\]
\[\operatorname{rank}A+\operatorname{nullity}A=3\]
Since \[\operatorname{rank}A=2\]
we get \[2+\operatorname{nullity}A=3\]
so \[\operatorname{nullity}A=1\]
Checks
- Solve the homogeneous system \(A\mathbf x=\mathbf 0\), not a system with a non-zero right-hand side.
- In rank-nullity, count columns of \(A\), not rows.
- Distinguish the zero vector from the zero-dimensional null space.
- Check each proposed null-space basis vector by substituting it into \(A\mathbf x=\mathbf 0\).