The discussion about the rank and nullity can be extended to linear transformations.
Let T: V → W be a linear transformation between vector spaces of finite dimensions. Then
rankT = dim(rangeT), nullityT = dim(kernelT).
By identifying abstract vector spaces with the euclidean ones, the properties about the rank and the nullity of a matrix can be translated to linear transformations.
Let T: V → W be a linear transformation between finite dimensional vector spaces. Then
The following is the characterization of invertibility in terms of the rank and nullity.
Let T: V → W be a linear transformation and dimV = dimW = n. Then the following are equivalent.
Example In an earlier example, we studied the linear transformation
|T(X) = AX - XA: M(2, 2) → M(2, 2), where A = [||1||2||]|
and found kernelT consists of two matrices. Therefore
nullityT = 2, rankT = dimM(2, 2) - 2 = 2.
In particular, T is not invertible.
Example The kernel of the linear transformation D(p(t)) = p'(t): Pn → Pn-1 consists of the constant polynomials. Therefore nullityD = 1 and rankD = dimPn - nullityD = n - 1 = dimPn-1. The computation on the rank tells us that D is onto.