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It follows that the null space of A is the orthogonal complement to the row space. 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 fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations: Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
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.
The nonzero rows of the echelon form are a basis for the row space of A. See the article on row space for an example. If we instead put the matrix A into reduced row echelon form, then the resulting basis for the row space is uniquely determined. This provides an algorithm for checking whether two row spaces are equal and, by extension, whether ...
Examples of vector spaces; Linear map. Shear mapping or Galilean transformation; Squeeze mapping or Lorentz transformation; Linear subspace. Row and column spaces; Column space; Row space; Cyclic subspace; Null space, nullity; Rank–nullity theorem; Nullity theorem; Dual space. Linear function; Linear functional; Category of vector spaces
The bilinear form used in Minkowski space determines a pseudo-Euclidean space of events. [5] The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal .
Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.
The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example, the counter-clockwise rotation matrix from above becomes: [ cos θ − sin θ 0 sin θ cos θ 0 0 0 1 ...