### 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

• rankT = rankT*.
• rankT + nullity(T) = dimV.
• rankTdimV and dimW.
• rankT = dimWT is onto.
• rankT = dimVT is one-to-one.

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.

• T is invertible.
• rangeT = W.
• rankT = n.
• kernelT = {0}.
• nullityT = 0.

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.