Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Linear Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search