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2. Transformation matrix - Wikipedia

   en.wikipedia.org/wiki/Transformation_matrix

   In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .

3. Proofs involving ordinary least squares - Wikipedia

   en.wikipedia.org/wiki/Proofs_involving_ordinary...

   Recall that M = I − P where P is the projection onto linear space spanned by columns of matrix X. By properties of a projection matrix, it has p = rank(X) eigenvalues equal to 1, and all other eigenvalues are equal to 0. Trace of a matrix is equal to the sum of its characteristic values, thus tr(P) = p, and tr(M) = n − p. Therefore,

4. Vectorization (mathematics) - Wikipedia

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

   The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, vec ⁡ ( A B C ) = ( C T ⊗ A ) vec ⁡ ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {T} }\otimes A)\operatorname {vec} (B)} for matrices A , B , and C of dimensions k ...

5. File:Linear transformation.pdf - Wikipedia

   en.wikipedia.org/wiki/File:Linear_transformation.pdf

   Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Pages for logged out editors learn more

6. Linear least squares - Wikipedia

   en.wikipedia.org/wiki/Linear_least_squares

   Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b' , where b' is the projection of b onto the column space of A .

7. Linear map - Wikipedia

   en.wikipedia.org/wiki/Linear_map

   A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to ...

8. Matrix equivalence - Wikipedia

   en.wikipedia.org/wiki/Matrix_equivalence

   In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.

9. Rotation matrix - Wikipedia

   en.wikipedia.org/wiki/Rotation_matrix

   Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...