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/
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