2.9 The Rank Theorem¶
We will repeat the following important theorem from the Margalit textbook.
- Rank of a matrix \(A\)
The dimension of the column space of \(A\) is the *rank+ of \(A\).
- Nullity of a matrix \(A\)
The dimension of the null space of \(A\) is the nullity of \(A\).
Rank Theorem The rank of matrix \(A\) plus the nullity of \(A\) equals the number of columns in \(A\). If \(A\) is an \(m\times n\) matrix, then
\[\text{rank}+\text{nullity}=n\]
Example: Rank of Matrix¶
M = [-4 3 25 11 ; -12 7 81 35 ; -2 2 11 5 ]
rref(M)
M =
-4 3 25 11
-12 7 81 35
-2 2 11 5
ans =
1.0000 0 -8.5000 -3.5000
0 1.0000 -3.0000 -1.0000
0 0 0 0
The matrix \(M\) has two pivots, so it’s column space is a 2-dimensional subspace of \(\mathbb R^3\) and it’s rank equals 2. There are two free variables, so the null space is a 2-dimensonial subspace of \(\mathbb R^4\). The solution vector
\[\begin{split}\vec x = \left[\begin{array}{r}x_1\\x_2\\x_3\\x_4\end{array}\right]\end{split}\]
has four components. Since there are two free variables, a basis for the null space will require two vectors, and hence the nullity of \(A\) also equals 2.