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2. 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 ...

3. Row and column spaces - Wikipedia

   en.wikipedia.org/wiki/Row_and_column_spaces

   For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space. The dimension of the null space is called the nullity of the matrix, and is related to the rank by the following equation:

4. Rank (linear algebra) - Wikipedia

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

   A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank. The rank deficiency of a matrix is the difference between the lesser of the number of rows and columns, and ...

5. Kernel (linear algebra) - Wikipedia

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

   In the case where V is finite-dimensional, this implies the rank–nullity theorem: ⁡ (⁡) + ⁡ (⁡) = ⁡ (). where the term rank refers to the dimension of the image of L, ⁡ (⁡), while nullity refers to the dimension of the kernel of L, ⁡ (⁡). [4] That is, ⁡ = ⁡ (⁡) ⁡ = ⁡ (⁡), so that the rank–nullity theorem can be ...

6. Jordan normal form - Wikipedia

   en.wikipedia.org/wiki/Jordan_normal_form

   In other words, map the set of matrix conjugacy classes injectively back into the initial set of matrices so that the image of this embedding—the set of all normal matrices, has the lowest possible degree—it is a union of shifted linear subspaces. It was solved for algebraically closed fields by Peteris Daugulis. [17]

7. Quotient space (linear algebra) - Wikipedia

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

   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). The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).

8. Rank (graph theory) - Wikipedia

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

   Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph. [2] Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti ...

9. Word problem (mathematics education) - Wikipedia

   en.wikipedia.org/wiki/Word_problem_(mathematics...

   Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.