I am having trouble understanding the relatioship between rows and columns of a matrix.Say, the following homogeneous system has a nontrivial solution. 3 x 1 + 5 x 2 − 4 x 3 = 0 − 3 x 1 − 2 x 2 + 4 x 3 = 0 6 x 1 + x 2 − 8 x 3 = 0 Let A be the coefficient matrix and row reduce [A0] to row-echelon form: [ 3 5 − 4 0 − 3 − 2 4 0 6 1 − 8 0 ] → [ 3 5 − 4 0 0 3 0 0 0 0 0 0 ] a 1 a 2 a 3Here, we see x 3 is a free variable and thus we can say 3rd column, a 3 , is in span ( a 1 , a 2 )But what does it mean for an echelon form of a matrix to have a row of 0's?Does that mean 3rd row can be generated by 1st &amp; 2nd rows?just like 3rd column can be generated by 1st &amp; 2nd columns?And this raises another question for me, why do we mostly focus on columns of a matrix?because I get the impression that ,for vectors and other concepts, our only concern iswhether the columns span R n or the columns are linearly independent and so on.I thought linear algebra is all about solving a system of linear equations,and linear equations are rows of a matrix, thus i think it'd be logical to focus more on rows than columns. But why?

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I am having trouble understanding the relatioship between rows and columns of a matrix.Say, the following homogeneous system has a nontrivial solution.  3      x    1    +  5      x    2    −  4      x    3    =  0    −  3      x    1    −  2      x    2    +  4      x    3    =  0    6      x    1    +      x    2    −  8      x    3    =  0  Let A be the coefficient matrix and row reduce [A0] to row-echelon form:      [                            3                          5                          −          4                          0                                      −          3                          −          2                          4                          0                                      6                          1                          −          8                          0                      ]    →      [                            3                          5                          −          4                          0                                      0                          3                          0                          0                                      0                          0                          0                          0                      ]      a  1    a  2      a  3Here, we see         x    3   is a free variable and thus we can say 3rd column,        a    3  , is in   span  (      a    1    ,      a    2    )But what does it mean for an echelon form of a matrix to have a row of 0&#039;s?Does that mean 3rd row can be generated by 1st &amp;amp; 2nd rows?just like 3rd column can be generated by 1st &amp;amp; 2nd columns?And this raises another question for me, why do we mostly focus on columns of a matrix?because I get the impression that ,for vectors and other concepts, our only concern iswhether the columns span             R        n   or the columns are linearly independent and so on.I thought linear algebra is all about solving a system of linear equations,and linear equations are rows of a matrix, thus i think it&#039;d be logical to focus more on rows than columns. But why?

humanistex3 · Accepted Answer

Having a row of 0&#039;s in the row-echelon form means that we were able to write the third row of A as a linear combination of the second and first rows. As it so happens for square matrices, this is true precisely when we can write the columns as a linear combination of each other (that is, when the columns are not linearly independent). If you further reduce this to reduced row-echelon form, you get      [                            1                          0                          −          4                      /                    3                          0                                      0                          1                          0                          0                                      0                          0                          0                          0                      ]  Because the third column lacks a pivot,       X    3   is our free variable, which means that we can write       a    3   as a linear combination of the other two columns.There&#039;s a very good reason for focusing on the columns of a matrix. This comes out of using A as a linear transformation, where the &quot;column space&quot; gives us the &quot;range&quot; of the function   f  (            x      →        )  =  A            x      →       .

I am having trouble understanding the relatioship between rows and columns of a matrix. Say, the fo

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