Linear combinations of the basis vectors

Using the basis vectors for as a starting point, we can actually build every vector in two-dimensional space, simply by adding scaled combinations of and . These scaled combinations are called linear combinations.

Example: Given vector , it “moves” units in the horizontal direction, or . It also “moves” units in the vectical direction, or .

So we could write a linear combination that expresses the vector, where we scale by , and scale by .

Which means we can define a new notation to express a vector: