1.1. Linear transformations¶
1.1.1. Kernel and image¶
For some \(T \in \mathcal{L}(V, W)\) we associate two subspaces – the kernel and the image.
Definition: Kernel
Let \(T : V \rightarrow W\) be a linear transformation. The kernel of \(T\) is defined
\[\text{ker} \, T = \{ v \in V \ |\ T(v) = 0 \}\]
Definition: Image
Let \(T : V \rightarrow W\) be a linear transformation. The image of \(T\) is defined
\[\text{im} \, T = \{ T(v) \ |\ v \in V \}\]
1.1.2. Rank and nullity¶
Definition: Rank
The rank of a linear transformation \(T\) is the dimension of its image
\[\text{rank} \, T = \text{dim} \, \text{im} \, T.\]
Definition: Nullity
The nullity of a linear transformation \(T\) is the dimension of its kernel
\[\text{nullity} \, T = \text{dim} \, \text{ker} \, T.\]
Theorem: Rank-nullity
Let \(T: \mathcal{L}(V, W)\), then
\[\text{rank} \, T + \text{nullity} \, T = \text{dim} \, V.\]