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2. Rank (linear algebra) - Wikipedia

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

   In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4] Rank is thus a measure of the "nondegenerateness ...

3. Rank–nullity theorem - Wikipedia

   en.wikipedia.org/wiki/Rank–nullity_theorem

   Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and. the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...

4. Linear algebraic group - Wikipedia

   en.wikipedia.org/wiki/Linear_algebraic_group

   As a result, it makes sense to define the k-rank or split rank of a group G over k as the dimension of any maximal split torus in G over k. For any maximal torus T in a linear algebraic group G over a field k , Grothendieck showed that T k ¯ {\displaystyle T_{\overline {k}}} is a maximal torus in G k ¯ {\displaystyle G_{\overline {k}}} . [ 12 ]

5. Sherman–Morrison formula - Wikipedia

   en.wikipedia.org/wiki/Sherman–Morrison_formula

   In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a " rank -1 update" to a matrix whose inverse has previously been computed. [1][2][3] That is, given an invertible matrix and the outer product of vectors and the formula cheaply computes an updated matrix inverse.

6. Cramer's rule - Wikipedia

   en.wikipedia.org/wiki/Cramer's_rule

   Cramer's rule. In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one ...

7. Vandermonde matrix - Wikipedia

   en.wikipedia.org/wiki/Vandermonde_matrix

   Vandermonde matrix. In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix. with entries , the jth power of the number , for all zero-based indices and . [1]