Category:Definitions/Kernels of Linear Transformations
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This category contains definitions related to Kernels of Linear Transformations.
Related results can be found in Category:Kernels of Linear Transformations.
Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules.
Let $e_H$ be the identity of $H$.
The kernel of $\phi$ is defined as:
- $\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$
where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$.
In Vector Space
Let $\struct {\mathbf V, +, \times}$ be a vector space.
Let $\struct {\mathbf V', +, \times}$ be a vector space whose zero vector is $\mathbf 0'$.
Let $T: \mathbf V \to \mathbf V'$ be a linear transformation.
Then the kernel of $T$ is defined as:
- $\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$
Pages in category "Definitions/Kernels of Linear Transformations"
The following 4 pages are in this category, out of 4 total.