Math question?

In R^4 , let w be the subset of all vectors?


In R^4 , let w be the subset of all vectors








V= a1


a2


a3


a4


that satisfy a4-a3=a2-a1


(a) Show that W is a subspace of R4


(b) Show that





S= 1 0 1 0


0 , 1 , 1 , 0


0 0 1 1


-1 1 1 1








(c ) Find a subset of s that is a basic for w





(c) Express as a linear combination of the basic


obtain in part C


v = 0


4


2


6

Math question?
a) To prove that a subset of a vector space is a subspace, you need to prove that the subset is closed under vector addition and scalar multiplication. Suppose (a,b,c,d) and (w,x,y,z) are two elements of V. Then d-c=b-a and z-y=x-w. Now (a,b,c,d)+(w,x,y,z)=(a+w,b+x,c+y,d+z). Since z-y=x-w, we can add these terms to the former equation, which gives us d-c+z-y=b-a+x-w, which in turn gives us (d+z)-(c+y)=(b+x)-(a+w). Thus shows that our vector sum must always be in the subset, so V is closed under vector addition. For any scalar k in R, k(a,b,c,d)=(ka,kb,kc,kd), but since we know that d-c=b-a, we can multiply this entire equation by k to get kd-kc=kb-ka. Thus, V is closed under scalar multiplication, and hence is a subspace of R^4.





I would do the other parts too, but I've got to go to class. Maybe I'll edit this answer later.





Ben