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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
- On ΛⁿV, the induced map ΛⁿT is multiplication by det T Corollary
- The inverse of an invertible finite-dimensional endomorphism is a polynomial in that endomorphism Corollary
- Orientation of a finite-dimensional real vector space Definition
- Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface Definition
- The norm N_K/F and trace Tr_K/F of a finite field extension Definition
- A nonidentity projection of the plane has determinant zero and is not invertible Example
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero Theorem
- Field norm and trace agree with the determinant and trace of multiplication by an element Theorem
- For endomorphisms S and T of one finite-dimensional vector space, det(ST)=det(S)det(T) Theorem
- Gauss’s Theorema Egregium Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then det(T)=∏_i<nλᵢ: determinant is the product of the eigenvalues counted with algebraic multiplicity Theorem
- Norm is multiplicative, trace is F-linear, and both are transitive in towers 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 · two levels
28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed. (standard reference, not scraped)