### 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`: `V` → `W` be a linear transformation between vector spaces of finite dimensions. Then

`rank`**T** = `dim`(`range`**T**), `nullity`**T** = `dim`(`kernel`**T**).

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

`rank`**T** = `rank`**T**^{*}.
`rank`**T** + `nullity`(`T`) = `dim`**V**.
`rank`**T**≤ `dim`**V** and `dim`**W**.
`rank`**T** = `dim`**W** ⇔ `T` is onto.
`rank`**T** = `dim`**V** ⇔ `T` is one-to-one.

The following is the characterization of invertibility in terms of the rank and nullity.

Let `T`: `V` → `W` be a linear transformation and dim`V` = dim`W` = `n`. Then the following are equivalent.

`T` is invertible.
`range`**T** = `W`.
`rank`**T** = `n`.
`kernel`**T** = {**0**}.
`nullity`**T** = 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 `kernel`**T** consists of two matrices. Therefore

`nullity`**T** = 2, `rank`**T** = `dim``M`(2, 2) - 2 = 2.

In particular, `T` is not invertible.

Example The kernel of the linear transformation `D`(`p`(`t`)) = `p`'(`t`): **P**_{n} → `P`_{n-1} consists of the constant polynomials. Therefore `nullity`**D** = 1 and `rank`**D** = `dim`**P**_{n} - `nullity`**D** = `n` - 1 = `dim`**P**_{n-1}. The computation on the rank tells us that `D` is onto.