ok, another problem suggested by my prof.
the vectors $u_1 = (1,1,1,1)$, $u_2 = (0,1,1,1)$, $u_3 = (0,0,1,1)$, $u_4 = (0,0,0,1)$, are a basis for $F^4$.
Find a unique representation of an arbitrary vector $(a_1, a_2, a_3, a_4)$ as a linear combination of $u_1, u_2, u_3, u_4$.
OK, as I understand it I have to find some scalar coefficients such that
$(a_1, a_2, a_3, a_4)$ = $c_n(u_1 + u_2 + u_3 + u_4)$ = $c_1(1,1,1,1) + c_2(0,1,1,1) + c_3(0,0,1,1) +c_4(0,0,0,1)$, correct?
assuming that is the case, though, I get stuck. if I just add the u-vectors I get (1, 2, 3, 4) but I am unsure what to do after that. I feel like this should be simple but I have been so bloody lost in this class..
Best Answer
You need $a_1u_1$ or else you won't get the right first coefficient. The only way to get $a_2$ in the second coefficient is by taking $u_2$, but if you take $a_2u_2$ you'll get $a_1+a_2$ in the second coefficient, which is too many. Hence, you must take $(a_2-a_1)u_2$, which will give you just $a_2$ in the second coefficient when added to $a_1$ from the $u_1$ previously.
Continuing in this way you will get $$a_1u_1+(a_2-a_1)u_2+(a_3-a_2)u_3+(a_4-a_3)u_4$$