A linear combination of a collection of vectors is any combination of those vectors you can make by taking sums and scalar multiples of them. Here's an example.

In this learning object, you're going to look at the vectors you can get as linear combinations of other vectors. Specifically, you'll learn

1. What the span of a set of vectors is,
2. Which vectors in 2-space or in 3-space can be spanned by a set of two vectors,
3. Which vectors in 2-space or in 3-space can be be spanned by a set of three or more vectors.

 

In the diagram below, the purple vector is a linear combination of the red and blue vectors. Use the sliders to show different linear combinations of the vectors. Which vectors do you get if one of the coefficients is 0?

For more than two vectors, the idea is the same, though not as easy to visualize. For example, the vector 2u + 3v - 5w is a linear combination of the vectors u, v and w. So is the vector

4(3w + 2u - v) - 2(v + 4u),

which simplifies to 12w - 6v. In general, any linear combination of u, v and w simplifies to a vector of the form au + bv + cw for some scalars a, b and c (some of which may be 0).

You can also change the red and blue vectors in the diagram by dragging their heads. What happens if the red and blue vectors are parallel?