# Finding a coordinate system for the kernel and nullity of a linear map

Let $T : C^{\infty}(\mathbb{R}) \rightarrow C^{\infty}(\mathbb{R})$ be a linear map, which is defined by $T = \frac{d^2}{d x^2} - 3\frac{d}{dx} + 2 \text{id}$, that is, a map which takes a function $f(x)$ and maps it to $$f''(x) - 3f'(x) + 2f(x)$$

Now my exercise asks me to find coordinate systems for the kernel of $T$ and for the nullity of $T$.

The kernel is the vectors in $C^{\infty}(\mathbb{R})$ which maps to $0$, and thus, it is exactly the functions on the form $\alpha e^{2x} + \beta e^{x}$, since that is the general solution to the differential equation $f''(x) - 3f'(x) + 2f(x) = 0$. To get a coordinate system for the kernel, I take the linear map from $\mathbb{R}^2 \rightarrow \text{ker} T$ associated with the matrix $$\begin{pmatrix} e^{2x} & e^{x}\\ \end{pmatrix}$$

This map is both linear and bijective, thus an isomorphism of vector spaces.

My problem is with the second part. By my notes, the nullity of the map $T$ is the dimension of its kernel. In this case, the dimension of the kernel is 2. How do I define a coordinate system on the number 2? Have I misunderstood something?

The exact wording of the exercise:

Let $T : C^{\infty}(\mathbb{R}) \rightarrow C^{\infty}(\mathbb{R})$ be the linear map defined by $T = \frac{d^2}{d x^2} - 3\frac{d}{dx} + 2 \text{id}$. Find a co-ordinate system for $\text{ker} T$ and $\text{null} T$.

-
I'm with you. I thought, maybe "nullity" was supposed to be "nullspace", but then that's the same as the kernel, so it seems a bit redundant. – Gerry Myerson Aug 20 '12 at 5:36
The kernel and nullspace of a linear transformation are exactly the same thing. – user38268 Aug 20 '12 at 5:40
The exact wording of the exercise has a greater-than symbol between the 2 and the id? – Gerry Myerson Aug 20 '12 at 5:46
@Gerry No, that was introduced by the editor when trying to make a citation. – utdiscant Aug 20 '12 at 5:48

The intended meaning of the question is "Find a coordinate system for $\text{ker}\ T$, and find $\text{null}\ T$".
So there is only one question remaining: what is null $T$? – Fabian Aug 20 '12 at 6:31
I think you are right. @Fabian: null $T$ is the dimension of the kernel, thus 2. – utdiscant Aug 20 '12 at 6:45