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The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and on the zero space
Definition
Let be a linear operator on a finite-dimensional vector space over a field (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis).
If , choose an ordered basis and define
where is the coordinate matrix of (Coordinate columns and matrices of linear maps relative to ordered bases) and the determinant on the right is the matrix determinant (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix). If , define .
The positive-dimensional definition is justified by The determinant of a linear operator is independent of the chosen ordered basis ↗, which proves that the value does not depend on the chosen ordered basis.
Depends on
- Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
- Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
Used by
- A nonidentity projection of the plane has determinant zero and is not invertible Example
- A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero Theorem
- For endomorphisms S and T of one finite-dimensional vector space, det(ST)=det(S)det(T) Theorem
- On a positive-dimensional space, det(T) is the unique scalar by which T scales every alternating top-degree form Theorem
- The determinant of a linear operator is independent of the chosen ordered basis Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 87 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed. (standard reference, not scraped)