Definition:Kernel of Linear Transformation/Vector Space
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This page is about Kernel of Linear Transformation on Vector Space in the context of linear algebra. For other uses, see kernel.
Definition
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:
- $\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$
Also known as
The kernel of a linear transformation $T$ on a vector space is also known as the null space on $T$.
Also denoted as
The notation $\map {\mathrm {Ker} } \phi$ can sometimes be seen for the kernel of $\phi$.
It can also be presented as $\ker \phi$ or $\operatorname {Ker} \phi$, that is, without the parenthesis indicating a mapping.
Also see
- Results about kernels of linear transformations can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.16$: Definitions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): kernel
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): null space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): kernel
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): null space
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations