## anonymous 5 years ago Let D={α1,α2} and E={β1,β2} be two bases for the plane, and suppose that β1=5(α1)-3(α2) and β2=-3(α1) + 2(α2)

1. anonymous

a) express α1 in the form α1=b1β1 + b2β2 and find an expression for α2

2. anonymous

seems like you need to find a vector that satisfies D....hmm

3. anonymous

are you getting $\alpha1 = b1\left[ 5(a1) - 3(a2) \right] + b2\left[ -3(a1)+2(a2) \right]$? Then for $\alpha2$ it might be similar

4. anonymous

the answer for $\alpha _{1}=2\beta _{1}+3\beta _{2}$

5. anonymous

though I don't know how to get to the answer

6. anonymous

$\alpha _{2}=3\beta _{1}+5\beta _{2}$ I have a difficult time understanding most of basis and S-coordinates

7. anonymous

Can anyone help me out?

8. anonymous

so is this a set?

9. anonymous

yup D and E (which are really S and T) are two sets or basis for the plane

10. anonymous

how would you construct the matrix? so that I can eliminate $\alpha _{2}$

11. anonymous

sorry, basis and s-coordinates are still really shaky for me

12. anonymous

ok here its simple...you dont need matrix

13. anonymous

ok here its simple...you dont need matrix

14. anonymous

$\beta_{1}= 5\alpha_{1} - 3\alpha_{2}$ (1) $\beta_{2}= -3\alpha_{1} + 2\alpha_{2}$ (2) if you multiply (1) by 2, and if you multiply (2) by 3 you get $\2beta_{1}= 10\alpha_{1} - 6\alpha_{2}$ $\3beta_{2}= -9\alpha_{1} + 6\alpha_{2}$ adding the above two equations, you eliminate the parameter $\alpha_{2}$ and it becomes $\2beta_{1} + \3beta_{2}= \alpha_{1}$

15. anonymous

oh awesome, ok thanks a lot! I'm guessing for $\alpha _{2}$ we eliminate (alpha)1

16. anonymous

yeah exact same way...sets are usually simple to solve if you know the algebra "tricks"

17. anonymous

you dont need a lot to solve them i mean...they seem difficult but it usually is very easy

18. anonymous

cool, hey since I have you here would you know how to conclude that A is non-singular given the equation $In=-1/2[A ^{2}-7A+In]A$

19. anonymous

nvm, i think i got it, A[-1/2A^2+7/2A-1/2IN]=In....where [] equal A^-1?

20. anonymous

yeah if you find the correct inverse... check to be sure that that matrix is infact an inverse

21. anonymous

How would you do that? A isn't defined

22. anonymous

hmmm....its trivial...you let it be arbitrary...look at the determinant for such arbitrary A

23. anonymous

So let A be 2x2 matrix [a,b][c,d] and solve?

24. anonymous

you can. if its nonzero then A is non-singular

25. anonymous

K thanks, going back to the basis question, if the S coordinates of point P are (-9, 3), what are it's T coordinates?

26. anonymous

27. anonymous

yeah you can use either system used to solve to find the T coord.

28. anonymous

this is hard but im here to help