Adding vectors (right side): vector a (blue) is added to vector b (red) by placing them as shown on the right. The sum, a+b, is black. Left side: the sum of two vectors makes the diagonal of a parallelogram.

A vector space is a collection of mathematical objects called vectors, along with some operations you can do on them. Two operations are defined in a vector space: addition of two vectors and multiplication of a vector with a scalar. These operations can change the size of a vector and the direction it points to. The most important thing to understand is that after you do the addition or multiplication, the result is still in the vector space; you have not changed the vector in a way that makes it not a vector anymore. A vector space is often represented using symbols such as \displaystyle{ U }, \displaystyle{ V } and \displaystyle{ W }.