Question

linear algebra: span?

-a+b-c+d=0
8a+4b+2c+d=0

how does this set of equations tell us that c=-b-3a and d=-2a-2b? I tried to to row reduce it but I didn't end up with the answer..

Answer 1

Here is my row reduction. What I kept in mind was to create an \(I_2\) submatrix in the \(c\) and \(d\) columns.\[
\begin{align}\left[\begin{array}{cccc|c} -1&1&-1&1&0\\8&4&2&1&0\end{array}\right]&&\text{given system}\\
\left[\begin{array}{cccc|c} 1&-1&1&-1&0\\8&4&2&1&0\end{array}\right] && R_1\mapsto -R_1\\
\left[\begin{array}{cccc|c} 1&-1&1&-1&0\\6&6&0&3&0\end{array}\right] && R_2\mapsto -R_2-2R_1\\
\left[\begin{array}{cccc|c} 1&-1&1&-1&0\\2&2&0&1&0\end{array}\right] && R_2\mapsto \frac{1}{3}R_2\\
\left[\begin{array}{cccc|c} 3&1&1&0&0\\2&2&0&1&0\end{array}\right] && R_1\mapsto R_1+R_2
\end{align}\]We can now write this as equations.\[\begin{align}3a+b+c&=0\\2a+2b+d&=0\end{align}\]This is just an alternate form of what you were to show. Just as a note: it's easier to do this by just simply substituting algebraically.

Answer 2

Thanks so much!