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2. Change of basis - Wikipedia

   en.wikipedia.org/wiki/Change_of_basis

   It is represented on "old" bases of V and W by a m×n matrix M. A change of bases is defined by an m×m change-of-basis matrix P for V, and an n×n change-of-basis matrix Q for W. On the "new" bases, the matrix of T is . This is a straightforward consequence of the change-of-basis formula.

3. Dual basis in a field extension - Wikipedia

   en.wikipedia.org/wiki/Dual_basis_in_a_field...

   Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication ...

4. Matrix multiplication algorithm - Wikipedia

   en.wikipedia.org/wiki/Matrix_multiplication...

   The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:

5. Hypercomplex number - Wikipedia

   en.wikipedia.org/wiki/Hypercomplex_number

   Putting aside the bases which contain an element e i such that e i 2 = 0 (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cl p,q (), indicating that the algebra is constructed from p simple basis elements with e i 2 = +1, q with e i 2 = −1, and ...

6. Matrix multiplication - Wikipedia

   en.wikipedia.org/wiki/Matrix_multiplication

   The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, although the tropical semiring is also a common choice for graph shortest path problems. [15]

7. Computational complexity of mathematical operations - Wikipedia

   en.wikipedia.org/wiki/Computational_complexity...

   Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.

8. Matrix (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Matrix_(mathematics)

   If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix analysis: the vector's components are the light ray's slope and its distance from the optical axis, while ...

9. Basis (linear algebra) - Wikipedia

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

   Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. [1] In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.