Thread: finding basis vecto space subspace

(9,0,8)

Originally Posted by jecowap

assuming that S and T are subspaces of V3(C) generated by
<(1,1,0),(i,1+i,1),(1+i,1+i,0)> and <(1,0,1),(i,-i,0),(0,i,i)> respectively
determine C-bases for S, T, SnT, S+T

Finding bases for S and T is easy: if the given sets are independent, then they are bases. If not, then you can solve for one (or two) of the vectors in terms of the others.

For example, to see if <(1,0,1), (i,,-i,0), (0,i,i)> is an independent set, look at a(1,0,1)+ b(i,-i,0)+ c(0,i,i)= (0,0,0). That is the same as (a+ ib, -ib+ ic, a+ ic)= (0,0,0) so we have the three equations a+ ib= 0, -ib+ ic= 0, a+ ic= 0. The middle equation is the same as ib= ic or b= c. The first and second equations, both reduce to a= -ib. If, for example, we take b= 1, then a= -i and c= 1 so -i(1,0,1)+ (i, -i, 0)+ (0,i,i)= (-i+i, -i+i, -i+i)= (0,0,0). That set is not independent and we can now see that (i,-i,0)= i(1,0,1)- (0,i,i) so we don't need the vector (i,-i,0)- it can be written using the other two. So <1,0,1),(0,i,i)> is enough for a basis. (Actually, any two of the three vectors forms a basis.)

Originally Posted by HallsofIvy

Finding bases for S and T is easy: if the given sets are independent, then they are bases. If not, then you can solve for one (or two) of the vectors in terms of the others.

For example, to see if <(1,0,1), (i,,-i,0), (0,i,i)> is an independent set, look at a(1,0,1)+ b(i,-i,0)+ c(0,i,i)= (0,0,0). That is the same as (a+ ib, -ib+ ic, a+ ic)= (0,0,0) so we have the three equations a+ ib= 0, -ib+ ic= 0, a+ ic= 0. The middle equation is the same as ib= ic or b= c. The first and second equations, both reduce to a= -ib. If, for example, we take b= 1, then a= -i and c= 1 so -i(1,0,1)+ (i, -i, 0)+ (0,i,i)= (-i+i, -i+i, -i+i)= (0,0,0). That set is not independent and we can now see that (i,-i,0)= i(1,0,1)- (0,i,i) so we don't need the vector (i,-i,0)- it can be written using the other two. So <1,0,1),(0,i,i)> is enough for a basis. (Actually, any two of the three vectors forms a basis.)

thanks so much
but for the other set i get to the abc point and the first and second are equal (a+c+ci=0) and b=0
could you possibly help me further with where to go from this point?