## Definition

Let $$T:V\rightarrow W$$ be a linear transformation where $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. The kernel of $$T$$, denoted by $$\ker(T)$$, is the set of vectors from $$V$$ that gets mapped to the zero vector in $$W$$; that is, $\ker(T) = \{ v \in V : T(v) = 0_W\}.$

### Examples

1. Let $$T$$ be given by $$T(x) = Ax$$ for some $$A \in \mathbb{R}^{m \times n}$$. By definition, the kernel of $$T$$ is given by the set of $$x$$ such that $$T(x) = 0$$. But $$T(x) = 0$$ precisely when $$Ax = 0$$. Therefore, $$\ker(T) = N(A)$$, the nullspace of $$A$$.

2. Let $$T$$ be a linear transformation from $$P_2$$ to $$\mathbb{R}^2$$ given by $$T(ax^2 + bx + c) = \begin{bmatrix} a+3c \\ a-c\end{bmatrix}$$. The kernel of $$T$$ is the set of polynomials $$ax^2+bx+c$$ such that $$\begin{bmatrix} a+3c \\a - c\end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}$$. Solving for $$a$$ and $$c$$, we get $$a = c = 0$$. So $$\ker(T) = \{ bx : b \in \mathbb{R}\}$$.

### Application

Let $$T:V\rightarrow W$$ be a linear transformation. Suppose that you are asked to find all solutions to $$T(x) = b$$ for some $$b \in W$$. If you have found one solution, say $$\tilde{x}$$, then the set of all solutions is given by $$\{\tilde{x} + \phi : \phi \in \ker(T)\}$$. In other words, knowing a single solution and a description of the kernel of $$T$$ tells you all the solutions to $$T(x) = b$$.

When $$T(x)$$ is given by $$Ax$$, this amounts to saying that to specify all solutions to $$Ax = b$$, we just need one particular solution and a description of the nullspace of $$A$$.

## Exercises

Give the kernel for each of the following linear transformations.

1. $$T(ax+b) = a-b$$ where $$a,b\in \mathbb{R}$$.

2. $$T(x) = Ax$$ where $$x = \begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix}$$ and $$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$.