anonymous
  • anonymous
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)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
a) express α1 in the form α1=b1β1 + b2β2 and find an expression for α2
anonymous
  • anonymous
seems like you need to find a vector that satisfies D....hmm
anonymous
  • 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

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anonymous
  • anonymous
the answer for \[\alpha _{1}=2\beta _{1}+3\beta _{2}\]
anonymous
  • anonymous
though I don't know how to get to the answer
anonymous
  • anonymous
\[\alpha _{2}=3\beta _{1}+5\beta _{2}\] I have a difficult time understanding most of basis and S-coordinates
anonymous
  • anonymous
Can anyone help me out?
anonymous
  • anonymous
so is this a set?
anonymous
  • anonymous
yup D and E (which are really S and T) are two sets or basis for the plane
anonymous
  • anonymous
how would you construct the matrix? so that I can eliminate \[\alpha _{2}\]
anonymous
  • anonymous
sorry, basis and s-coordinates are still really shaky for me
anonymous
  • anonymous
ok here its simple...you dont need matrix
anonymous
  • anonymous
ok here its simple...you dont need matrix
anonymous
  • 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 \[\2beta_{1} + \3beta_{2}= \alpha_{1} \]
anonymous
  • anonymous
oh awesome, ok thanks a lot! I'm guessing for \[\alpha _{2}\] we eliminate (alpha)1
anonymous
  • anonymous
yeah exact same way...sets are usually simple to solve if you know the algebra "tricks"
anonymous
  • anonymous
you dont need a lot to solve them i mean...they seem difficult but it usually is very easy
anonymous
  • anonymous
cool, hey since I have you here would you know how to conclude that A is non-singular given the equation \[In=-1/2[A ^{2}-7A+In]A\]
anonymous
  • anonymous
nvm, i think i got it, A[-1/2A^2+7/2A-1/2IN]=In....where [] equal A^-1?
anonymous
  • anonymous
yeah if you find the correct inverse... check to be sure that that matrix is infact an inverse
anonymous
  • anonymous
How would you do that? A isn't defined
anonymous
  • anonymous
hmmm....its trivial...you let it be arbitrary...look at the determinant for such arbitrary A
anonymous
  • anonymous
So let A be 2x2 matrix [a,b][c,d] and solve?
anonymous
  • anonymous
you can. if its nonzero then A is non-singular
anonymous
  • 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?
anonymous
  • anonymous
you just plug them in to your answers from i), sorry
anonymous
  • anonymous
yeah you can use either system used to solve to find the T coord.
anonymous
  • anonymous
this is hard but im here to help

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