I've been given a very confusing homework problem that is as follows:
Let U be the set of all vectors u in $ℝ^4$ such that $2(u_1) + 3(u_3) - 2(u_4) = 0$ (i.e. U is the solution space of a given system).
(a) Find the dimension of U.
(b) Find a basis of U.
(c) Write U using the basis.
So firstly I'm not sure what $2(u_1) + 3(u_3) - 2(u_4) = 0$ . Is this vector the solution space of all other vectors in U?
If the dimension of a vector space Dim(U)=n then the dimension should be 4, no? Furthermore a basis of U should be a linear combination of any vector in the space, so would a linear combination of the given vector [2 0 3 -2] be sufficient?
I'm finding only abstractions of this problem in the book, my notes, and online.