In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.
In linear algebra, given a quotient map, the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel.
In fiber bundles, the relative dimension of the map is the dimension of the fiber.
More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel.
These are dual in that the inclusion of a subspace of codimension k dualizes to yield a quotient map of relative dimension k, and conversely.
The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fiber product.
Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.
Famous quotes containing the words relative and/or dimension:
“She went in there to muse on being rid
Of relative beneath the coffin lid.
No one was by. She stuck her tongue out; slid.”
—Gwendolyn Brooks (b. 1917)
“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a dimension where governments rose and fell as a result of arguments between two colonels’ wives.”
—Frances Fitzgerald (b. 1940)