kernel of a linear mapping

Let $T\colon V\to W$ be a linear mapping between vector spaces.

The set of all vectors in $V$ that $T$ maps to $0$ is called the kernel (or nullspace) of $T$ , and is denoted $\ker T$ . So$$ \ker T = \{\, x \in V\mid T(x)=0\,\}.$$

The kernel is a vector subspace of $V$ , and its dimension is called the nullity of $T$ .

The function $T$ is injective if and only if $\ker T=\{0\}$ (see the attached proof). In particular, if the dimensions of $V$ and $W$ are equal and finite, then $T$ is invertible if and only if $\ker T=\{0\}$ .

If $U$ is a vector subspace of $V$ , then we have$$ \ker T|_U = U \cap \ker T,$$ where $T|_U$ is the restriction of $T$ to $U$ .

When the linear mappings are given by means of matrices, the kernel of the matrix $A$ is$$ \ker A=\{\,x\in V \mid Ax=0\,\}.$$