Set of all linear combinations

Question

This is my first time posting a question to this forum (so bare with me). Struggling a bit on trying to understand this problem:

Let u and v be two vectors, what is the set of all linear combinations cu + dv for scalars c and d such that c + d = 1.

Would the set just be all real numbers?

Answer 1

{ cu + dv : c + d = 1 } = { cu + (1 - c)v : c in R } =
{ c(u - v) + v : c in R } is a one dimensional vector space,
a line passing through the ends of u (c = 1) and v (c = 0).

Answer 2

Linear combination of vectors where their coefficients add up to 1 is called a Convex combination. Here, since the coefficients are c & d and c+d=1 , we have cu+dv is a convex combination. Any convex combination of two points P1 and P2 generates a point P that lies on the line segment joining P1 and P2. Therefore, the set of all convex combinations of u and v is the set of all points on the line segment joining u and v. Now, the type of elements in the set depends on which vector space u and v belong to. If u and v belongs to the R2 space then their convex combination will also be in the same space. If u and v are on the real line (R) then cu+dv will also be a real number