1. anonymous

Let u and v be non-parallel vectors in R2, and let w be any vector in R2. Show for unique constants s and t, w = su + tv

2. anonymous

It might help to write what I've been trying: I separated the vectors into horizontal and vertical components to get|dw:1442812756247:dw|

3. anonymous

I got |dw:1442812854324:dw|

4. jim_thompson5910

I think I might have something, but I used matrices to get there. Would that work?

5. anonymous

Sure I've covered matrices in my class

6. DanJS

take me back to linear algebra and linear transforms please.. ..

7. DanJS

comb

8. jim_thompson5910

ok so I started off letting w = <x,y> be some vector in the plane of R2 I also let u = <a,b> v = <c,d> so that means w = s*u + t*v <x,y> = s*<a,b> + t*<c,d> <x,y> = <s*a,s*b> + <t*c,t*d> <x,y> = <s*a+t*c, s*b+t*d> x = s*a+t*c x = as + ct y = s*b+t*d y = bs + dt In this case, the scalars a,b,c,d are constant while s,t are variables. The system below as + ct = x bs + dt = y turns into this matrix equation $\Large \begin{bmatrix}a & c\\b & d\end{bmatrix} \begin{bmatrix}s\\t\end{bmatrix} = \begin{bmatrix}x\\y\end{bmatrix}$ a solution exists if and only if the matrix with a,b,c,d in it is invertible. Which only happens if the determinant of that matrix is nonzero Let's say u and v were parallel. That would mean a/c = b/d ad = bc ad - bc = 0 So if u and v were non-parallel, then ad - bc (which is the determinant of the a,b,c,d matrix) is nonzero

9. anonymous

This makes sense but how can you say that s and t and unique constants instead of variables?

10. jim_thompson5910

I made s,t variables to allow <x,y> to vary The idea is that you have a fixed set of 2 vectors. Then you can have those vectors generate the entire plane based on a unique s,t solution

11. anonymous

Oh I get it now it will be unique based on the vector.

12. jim_thompson5910

The uniqueness comes from the fact that the solution of the matrix equation is unique

13. anonymous

Well done, thanks a lot!

14. jim_thompson5910

you're welcome

15. DanJS

good one