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  2. Kernel (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Kernel_(linear_algebra)

    The left null space of A is the same as the kernel of A T. The left null space of A is the orthogonal complement to the column space of A, and is dual to the cokernel of the associated linear transformation. The kernel, the row space, the column space, and the left null space of A are the four fundamental subspaces associated with the matrix A.

  3. Row and column spaces - Wikipedia

    en.wikipedia.org/wiki/Row_and_column_spaces

    For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin. This provides a proof of the rank–nullity theorem (see dimension above). The row space and null space are two of the four fundamental subspaces associated with a matrix A (the other two being ...

  4. Codimension - Wikipedia

    en.wikipedia.org/wiki/Codimension

    More generally, if W is a linear subspace of a (possibly infinite dimensional) vector space V then the codimension of W in V is the dimension (possibly infinite) of the quotient space V/W, which is more abstractly known as the cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition

  5. Rank–nullity theorem - Wikipedia

    en.wikipedia.org/wiki/Rank–nullity_theorem

    The second proof [6] looks at the homogeneous system =, where is a with rank, and shows explicitly that there exists a set of linearly independent solutions that span the null space of . While the theorem requires that the domain of the linear map be finite-dimensional, there is no such assumption on the codomain.

  6. Metric signature - Wikipedia

    en.wikipedia.org/wiki/Metric_signature

    The number v (resp. p) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp. negative-definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrix g ab of the scalar product. Thus a nondegenerate scalar product has signature (v, p, 0), with v ...

  7. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    ⁠ For example, in the above example the null space is spanned by the last row of ⁠ ⁠ and the range is spanned by the first three columns of ⁠. As a consequence, the rank of ⁠ M {\displaystyle \mathbf {M} } ⁠ equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in Σ ...

  8. Quotient space (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Quotient_space_(linear...

    The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).

  9. Incidence matrix - Wikipedia

    en.wikipedia.org/wiki/Incidence_matrix

    The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.