Row equivalent意思
"Row equivalent" is a term used in linear algebra, specifically in the context of matrices. Two matrices are said to be "row equivalent" if one can be transformed into the other by a series of elementary row operations. These operations typically include swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.
For example, consider two matrices A and B:
A = [ \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} ]
B = [ \begin{bmatrix} 1 & 2 & 3 \ 0 & -1 & -2 \ 0 & 0 & 0 \end{bmatrix} ]
We can transform matrix A into matrix B using the following sequence of row operations:
- Swap row 1 and row 2 (to get a 0 in the top-left corner).
- Replace row 2 with (-4 times row 1) + row 2.
- Replace row 3 with (-7 times row 1) + row 3.
After these operations, matrix A becomes:
[ \begin{bmatrix} 1 & 2 & 3 \ 0 & -1 & -2 \ 0 & 0 & 0 \end{bmatrix} ]
which is equal to matrix B. Therefore, matrices A and B are row equivalent.
Row equivalence is a fundamental concept in the study of linear systems and is closely related to the row space and column space of a matrix, as well as to the concept of a reduced row echelon form (RREF). It is also used in the process of Gaussian elimination to solve systems of linear equations.