Free columns-combinations of columns?

[sarvesh0303]

In Gilbert Strang's book on linear algebra, he mentions that pivot columns are not combinations of earlier columns and that free columns are combinations of earlier columns. What columns is he referring to by earlier columns?

Also, what property distinguishes pivot variables and free variables that we choose the values for free variables for special solutions. Why don't we choose the values for the pivot variables and then find the free variables?

 

[SmilingDave]

A is earlier than B means A is to the left of B.

Because it does not always work, starting with the pivot variables. But starting with the free variables always works.

For example, let's say that after Gaussian elimination, you end up with x+y=6, and no equations at all for z and w. X is the pivot variable. y, z and w free. You could start with x, and say y is 6-x, but then what do you do about z and w? But if you do it the other way, then what you do to the free variables tells you all about the pivot variables, too.

On a deeper level, "starting with x" is another way of saying "let x be a free variable". This would correspond to interchanging the columns for x and y to make y pivot and x free.

 

[sarvesh0303]

Thanks a ton. Wow... you just made things so much clearer for me!