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Rank and nullity of a linear map with finite-dimensional domain
Definition
Let be a linear map whose domain is finite-dimensional over . Its nullity and rank are
Both dimensions are defined. The kernel is a linear subspace of the finite-dimensional space , so it is finite-dimensional by The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial and If and is a linear subspace of , then is finite-dimensional, , and if and only if . The image is finite-dimensional because Extending a basis of the kernel to a basis of the domain gives a basis of the image constructs a finite basis of it from a basis of . Thus rank and nullity are natural numbers, including when is the zero space.
Depends on
- Kernel and image of a linear map
- The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial
- Extending a basis of the kernel to a basis of the domain gives a basis of the image
- Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
- If $\dim_F V = n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, $\dim_F U \le n$, and $\dim_F U = n$ if and only if $U = V$
Used by
- The rank of a matrix equals the rank of the linear map x↦ Ax Corollary
- The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space Definition
- The rank of a derivative and constant-rank Euclidean maps Definition
- For a subspace U≤ Fⁿ, dim_F U^⊥=n-dim_F U, where U^⊥={x:⟨ x,u⟩=0 for all u∈ U} Lemma
- Kernel and rank sequences of powers stabilise once equality occurs Lemma
- Exact column-pivoted QR of a real or complex rank-r matrix has an invertible leading triangular block and zero trailing block Theorem
- Power ranks determine every nilpotent Jordan-block multiplicity Theorem
- Rank-nullity: dim_F V=nullityT+rankT Theorem
- Ranks of shifted powers determine Jordan form up to block order Theorem
Dependency tree · two levels
38 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
- UCLA Algebra Notes, rank and nullity (standard reference, not scraped)
- Axler, Linear Algebra Done Right, 4th ed., Chapter 3 (standard reference, not scraped)