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What is the difference, if any, between kernel and null space?

I previously understood the kernel to be of a linear map and the null space to be of a matrix: i.e., for any linear map $f : V \to W$,

$$ \ker(f) \cong \operatorname{null}(A), $$

where

  • $\cong$ represents isomorphism with respect to $+$ and $\cdot$, and
  • $A$ is the matrix of $f$ with respect to some source and target bases.

However, I took a class with a professor last year who used $\ker$ on matrices. Was that just an abuse of notation or have I had things mixed up all along?

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2 Answers 2

up vote 8 down vote accepted

The terminology "kernel" and "nullspace" refer to the same concept, in the context of vector spaces and linear transformations. It is more common in the literature to use the word nullspace when referring to a matrix and the word kernel when referring to an abstract linear transformation. However, using either word is valid. Note that a matrix is a linear transformation from one coordinate vector space to another. Additionally, the terminology "kernel" is used extensively to denote the analogous concept as for linear transformations for morphisms of various other algebraic structures, e.g. groups, rings, modules and in fact we have a definition of kernel in the very abstract context of abelian categories.

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The mapping could in general be a homomorphism of algebraic structures. What else do you have in mind? –  Manos Nov 12 '12 at 14:15

"Was that just an abuse of notation or have I had things mixed up all along?" Neither. Different courses/books will maintain/not maintain such a distinction. If a matrix represents some underlying linear transformation of a vector space, then the kernel of the matrix might mean the set of vectors sent to 0 by that transformation, or the set of lists of numbers (interpreted as vectors in $\mathbb{R}^n$ representing those vectors in a given basis, etc.

The context should make things clear and every claim about, say, dimensions of kernels/nullspaces should still hold despite the ambiguity.

As manos said, "kernel" is used more generally whereas "nullspace" is used essentially only in Linear Algebra.

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