math111_logo Rank and Nullity

3. Rank and nullity for linear transformation

The discussion about the rank and nullity can be extended to linear transformations.

Let T: VW 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: VW 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: VW 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 ]
3 4

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): PnPn-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.


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