Question

Anyone understand Problem Set 3.2, Problem #20? I'm having trouble understanding the reasons for the answer.

Suppose col 1 + col 3 + col 5 = 0 in a 4 x 5 matrix w/ 4 pivots. Which col is sure to have a pivot (and which var. is free)? What is the special solution? What is the nullspace?

Answer 1

Well, in any 4by5 matrix, the fifth column is certain to not have a pivot variable. That is because a row cannot have more than 1 pivot. So even, if every row in the 4by5 matrix contains a pivot, by the time you get to the fifth colum, they will all be used up.

As for the nullspace: We know that \[column1 + column3 + column5 = \left[\begin{matrix}0\\0\\0\\0\end{matrix}\right] \], so a basis for the nullspace nullspace would be any multiple, c, of that combination of columns with no other columns added: \[c\left[\begin{matrix}1\\0\\1\\0\\1\end{matrix}\right] \] Finally, special solutions are just any combination of the nullspace. I'll show you with an example why that works, but it is because the special solutions are in the null space, adding them to the particular solution does not change the result.

Hypothetically, the 4x5 matrix could be
\[\left[\begin{matrix}1&0&1&0&-2\\0&0&1&0&-1\\0&1&0&0&0\\0&0&0&1&0\end{matrix}\right] \] And if you multiply that by the special solution, you should get the zero matrix:
\[\left[\begin{matrix}1&0&1&0&-2\\0&0&1&0&-1\\0&1&0&0&0\\0&0&0&1&0\end{matrix}\right] \left[\begin{matrix}1\\0\\1\\0\\1\end{matrix}\right] =\left[\begin{matrix}0\\0\\0\\0\end{matrix}\right]\]

Answer 2

Thanks!!

Answer 3

Your welcome.