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Why does row rank equal column rank?

2From linear systems of equations to matrices

Suppose we have a linear system of equations:

It consists of mm equations in the variables  x1,,xnx_1, \dots, x_n. The aija_{ij} are coefficients in R\R.

For which values y1,,ymRy_1, \dots, y_m \in \R do these equations have a solution (x1,,xn)(x_1, \dots, x_n)? This question certainly depends on the number of equations and their relationship to each other.

Example: The system

does not always have a solution. For instance, it has no solution for y1=0y_1 = 0 and y2=1y_2 = 1, since the second equation is two times the first equation. More generally the same reasoning gives that there does not exist a solution whenever 2y1y22y_1\neq y_2. On the other hand it has a solution for y1=1y_1 = 1 and y2=2y_2=2 or more generally for 2y1=y22y_1=y_2, for example x1=y1x_1 = y_1 and x2=0x_2=0.

Writing a linear system of equations the way we wrote it above carries a lot of redundancy. For example the variables x1,,xnx_1, \dots, x_n appear in every row. To ease notation, we can use the matrix vector multiplication: We assemble the coefficients aija_{ij} into an (m×n)(m\times n)-matrix and write the xix_i’s and yjy_j’s as vectors.

By definition of matrix vector multiplication, we have yi=ai1x1+ainxny_i = a_{i1}x_1 + \dots a_{in}x_n for each i{1,,m}i \in \{1, \dots, m\}.

Our example turns into the following matrix vector multiplication


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