Let H be the subspace of R^4 given by
H = {(r, s, t, u)^T | r, s, t, u ∈ R, r − 2s + t + 3u = 0 and s + t − 4u = 0 }

Find a basis for H and determine dim H.

Question

Let H be the subspace of R^4 given by
H = {(r, s, t, u)^T | r, s, t, u ∈ R, r − 2s + t + 3u = 0 and s + t − 4u = 0 }

Find a basis for H and determine dim H.

anonymous · Answer

I don't know anything

turingtest · Answer

I'm afraid my linear algebra is a bit shoddy, but @Zarkon surely know what do to, though he remains silent.

turingtest · Answer

don't we just set up the augmented matrix and collect the s,t, and u parts as  separate vectors?

turingtest · Answer

$$\left[\begin{matrix}1&-2&1&3\0&1&1&-4\end{matrix}\right]=\binom 00$$$$\left[\begin{matrix}1&0&3&-5\0&1&1&-4\end{matrix}\right]=\binom 00$$$$x_1=-3t+5u$$$$x_2=-t+4ut$$$$x_3=t$$$$x_4=u$$$$\vec x=t\binom{-3}{-1}+u\binom54$$am I on the right track?

zarkon · Answer

the vectors in H are 4-tuples.  Any basis vector will also be a 4-tuple

turingtest · Answer

so$$\vec v_1=\left[\begin{matrix}1\0\-3\-5\end{matrix}\right]$$$$\vec v_2=\left[\begin{matrix}0\1\5\4\end{matrix}\right]$$???
(linear algebra review needed obviously)

zarkon · Answer

$$\left[\begin{matrix}1&0&3&-5\0&1&1&-4\end{matrix}\right]$$

which uses the variables r,s,t,u so t and u are free variables...call them x and y respectively 

then r=-3x+5y and s= -x+4y

then 

$$\left[\begin{matrix}r\s\t\u\end{matrix}\right]=\left[\begin{matrix}-3x+5y\-x+4y\x\y\end{matrix}\right]$$

$$=\left[\begin{matrix}-3\-1\1\0\end{matrix}\right]x+\left[\begin{matrix}5\4\0\1\end{matrix}\right]y$$

turingtest · Answer

oh, I meant a -3 in the first vector, but put a 5 instead
well, I was almost close... thanks Zarkon

turingtest · Answer

meant a -1*

anonymous · Answer

Thank you so much!