Is there a generalization unifying kernels and determinants?

[Metronome]

The kernel of a transformation can be viewed as the degree to which it fails to be injective. In linear algebra, the determinant can sort of be viewed as the degree to which a transformation fails to not be injective (0 determinant entails not injective, low determinant entails compressing the vector space a lot but not "enough" to lose dimensions, granted quantifying "enough" would involve infinity). Presumably, the Jacobian Determinant is the right way to extend this invocation of the determinant to nonlinear transformations.

Is there a generalization unifying these two concepts, either for linear transformations or generally? Squinting hard enough, it almost feels like they're trying to convey the same continuum of information, but alone they do not do so exhaustively; once you lose a dimension to a linear transformation, the determinant is 0, and it doesn't give you any further information about whether remaining dimensions were compressed, so there is room for a more general framework to provide more information.

 

[blamocur]

once you lose a dimension to a linear transformation, the determinant is 0, and it doesn't give you any further information about whether remaining dimensions were compressed,

Could you be talking about rank of a matrix by chance ?

 

[Metronome]

Could you be talking about rank of a matrix by chance ?

Nah, with the appropriate qualifications, the rank expresses the same information as the nullity, but it doesn't express anything in the realm of the information the determinant would express (in an injective transformation) regarding any compression or expansion of the vector space which fails to collapse whole dimensions.

 

[fresh_42]

The determinant only distinguishes between zero and not zero with respect to injectivity. It cannot measure a sort of degree of infectivity. However, the determinant is a (oriented) volume, the volume of the parallelepiped spanned by its column vectors. It is zero as soon as this parallelepiped is not of full dimension.

 

[Metronome]

The determinant only distinguishes between zero and not zero with respect to injectivity. It cannot measure a sort of degree of infectivity. However, the determinant is a (oriented) volume, the volume of the parallelepiped spanned by its column vectors. It is zero as soon as this parallelepiped is not of full dimension.

I mean, since the "sort of" allows a small amount of hand-waving, I think we can give it a "degree of injectivity" interpretation. Zero matrices shrink everything to the origin, so a large, finite amount of shrinking seems like a transformation "headed in that direction." Similarly for a transformation that rotates a basis vector "almost" into the span of the others. Certainly in any applied linear algebra context, these types of transformations would run into quantization constraints that look a lot like degrees of non-injectivity (i.e., losing some information shrinking an image represented via a finite number of pixels).

 

[fresh_42]

The determinant is either zero or it is not. There is no in-between. As soon as it is not zero, it is an isomorphism, particularly injective.

 

[Metronome]

The determinant is either zero or it is not. There is no in-between. As soon as it is not zero, it is an isomorphism, particularly injective.

The determinant isn't binary. It takes on real values, some of which are absolutely closer to zero than others.

 

[fresh_42]

The determinant isn't binary. It takes on real values, some of which are absolutely closer to zero than others.

That's the volume. Injective or not is binary.

 

[Metronome]

That's the volume. Injective or not is binary.

Maybe another way to put it is that a reduction in the number of dimensions (i.e., reducing volume to area) seems like it ought to be treated as a special kind of reduction in the volume.

 

[fresh_42]

Maybe another way to put it is that a reduction in the number of dimensions (i.e., reducing volume to area) seems like it ought to be treated as a special kind of reduction in the volume.

If we consider a linear map [imath] \varphi \, : \,V \longrightarrow V [/imath] then [imath] V\cong \operatorname{im}\varphi \oplus \operatorname{ker}\varphi . [/imath] The image [imath] \operatorname{im}\varphi \subseteq V [/imath] is a subspace and [imath] \varphi |_{\operatorname{im}\varphi } [/imath] is injective. Its size is determined by the dimension of [imath] \operatorname{ker}\varphi . [/imath] The determinant is not suited to consider this situation. The better framework to investigate the degree of injectivity, and I interpret it as [imath] \dim \operatorname{ker}\varphi [/imath] is the short exact sequence
[math] \{0\} \longrightarrow \operatorname{ker}\varphi \longrightarrow V \longrightarrow \operatorname{im}\varphi \longrightarrow \{0\} \qquad (^*)[/math]and the fact that it splits. I linked to the Wikipedia article as it is usually easier to read. A better description can be found on nLab.

The determinant is an oriented volume. An area in a three-dimensional space has zero height and its volume (width times length times height) is therefore zero. To get to the area, we have to consider the two-dimensional subspace in which the area becomes a positive volume (width times length times). The value of this volume depends on whether you measure in inches or centimeters. The reduction of the situation to lower dimensions is what the [imath] (^*) [/imath] expresses independent of your ruler.