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Since row operations can affect linear dependence relations of the row vectors, such a basis is instead found indirectly using the fact that the column space of A T is equal to the row space of A. Using the example matrix A above, find A T and reduce it to row echelon form:
The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A. A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Three proofs of this result are given in § Proofs that column rank = row rank, below.)
Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix. Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity. For symmetric Toeplitz matrices, there is the decomposition
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 rank of a matrix is the number of nonzero rows in its reduced row echelon form. If the ranks of the coefficient matrix and the augmented matrix are different, then the last non zero row has the form [ 0 … 0 ∣ 1 ] , {\displaystyle [0\ldots 0\mid 1],} corresponding to the equation 0 = 1 .
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of . Then C {\textstyle C} is obtained by removing from A {\textstyle A} all non- pivot columns (which can be determined by looking for columns in B {\textstyle B} which do not contain a pivot), and F {\textstyle F} is ...
Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } .
A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of linear equations is said to be in reduced row echelon form or in canonical form if its augmented matrix is in reduced row echelon form. The canonical form may be viewed as an explicit solution of the linear system.
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