Kernel

The kernel of a linear transformation is the subspace of the domain of the transformation that maps to the zero vector. More plainly:

Kernel ^formula

Let $T:U\to V$ be a linear transformation. Then the kernel (or nullspace) of $T$ is given by:

$$\text{ker}(T)=\{\vec{u}\in U|T(\vec{u})=\vec{0}\}$$

The kernel is a subspace of $U$. The nullity of $T$ is the dimension of the kernel:

$$\text{nullity}(T)=\text{dim}(\text{ker}(T))$$

Obtaining the kernel §

Let $T:U\to V$

1. We can find the transformation matrix of a linear transformation, $T$, given by:

2. Find the solution space of $[T]$. I.e. solve:

$$[T][\vec{v}]=[\vec{0}]\vec{v}\in U$$

3. Because the transformation matrix is written as a matrix of coordinate vectors, our final set of solutions, $K=\{{\vec{v}}_1\dots {\vec{v}}_n\}$ is not the kernel, because $\vec{v}$ is a coordinate vector, not the ‘full vector’
4. Simply multiply the coordinate vectors by the basis vectors:

$$\vec{v}=[\vec{v}{]}_{\mathcal{B}}\cdot \mathcal{B}$$

5. The kernel is given as a set of all these solution vectors, i.e. a subspace of $U$

$$\text{ker}(T)=\{{\vec{v}}_1,{\vec{v}}_2,\dots {\vec{v}}_n\}$$