
Finding a vector space
Hey, I have a question here that asks:
Find a basis of the vector space U.
U = {(x1, ..., X6) belongs to R6  x1 = 2x2, x3 = 7x6}
I realize the basis has to have four elements and that I have to find a set of generators but I do not know where to go from there. Any help would be appreciated. Thanks.

is, of course, 6 dimensional and, generally, each equation reduces the dimension by 1 so your subspace is 6 2= 4 dimensional and you need a basis containing 4 vectors.
Here's a good way of finding that basis. You are given that x1= 2x2 and x3= 7x6. That means that you are not "free" to choose x1 and x3 to be whatever you want they are fixed by the values of x2 and x6. But all the others are "free". Take those to be, one at a time, equal to 1 while the other "free variables" are 0.
For example, if x2= 1, x4= x5= x6= 0 (I have skipped x1 and x3 because they are not "free") then x1= 2x2= 2 and x3= 7x6= 0 so we have <2, 1, 0, 0, 0, 0> as one basis vector. If x4= 1, x2= x5= x6, then x1= 2x2= 0 and x3= 7x6= 0 so we have <0, 0, 0, 1, 0, 0> as another basis vector. If x5= 1, x2= x4= x6= 0, then x1= 2x2= 0 and x3= 7x6= 0 so we have <0, 0, 0, 0, 1, 0> as a third basis vector. Finally, if x6= 1, x2= x4= x5= 0, then x1= 2x2= 0 and x3= 7x6= 7 so we have <0, 0, 7, 0, 0, 1> as the fourth basis vector.