Linear Combinations of Vectors

Date: 10/23/2005 at 16:45:30
From: Sara
Subject: Linear Combinations

Can every vector in the xy plane be written as a linear combination 
of the vectors u = (1,4) and v = (-2,5)?  Can every vector in the xy
plane be written as a linear combination of the vectors u = (-4,-6)
and v = (10,15)? 

I'm confused about when you can write one vector as a linear 
combination of one other.  I think you can do it if they are parallel
and cannot if they are not collinear.  I think you can write one
vector as a linear combination of two others if they are not parallel.

Am I on the right track?

Date: 10/24/2005 at 13:53:50
From: Doctor George
Subject: Re: Linear Combinations

Hi Sara,

Thanks for writing to Doctor Math.

You have the right idea.  Look at it this way.  If w is a linear
combination of u and v then there are constants a and b such that

             w = au + bv

Now write separate equations for the x and y components.

             wx = a*ux + b*vx

             wy = a*uy + b*vy

This is just two equations with two unknowns, so we can solve for a
and b.  The solution will exist for any w unless u and v are parallel.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/