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Linear Combinations in 2-space

  1. Jan 16, 2016 #1
    1. The problem statement, all variables and given/known data

    In the xy-plane mark all nine of these linear combinations:

    ## α \lbrack 2, 1 \rbrack + β \lbrack 0, 1 \rbrack## with c = 0, 1, 2 and d = 0, 1, 2


    2. Relevant equations

    ANSWER:

    The nine combinations will lie on a lattice. If we took all whole numbers c and d, the lattice would
    lie over the whole plane.

    3. The attempt at a solution

    I think my biggest problem in this, is not actually knowing what the question is asking.

    I've tried plotting the linear combinations of me filling in α and β with the numbers listed for c & d.

    1. ## 0 \lbrack 2, 1 \rbrack + 0\lbrack 0, 1 \rbrack ## yields, ## \lbrack 0, 0 \rbrack ##

    2. ## 1 \lbrack 2, 1 \rbrack + 1\lbrack 0, 1 \rbrack ## yields, ## \lbrack 2, 2 \rbrack ##

    3. ## 2 \lbrack 2, 1 \rbrack + 2\lbrack 0, 1 \rbrack ## yields, ## \lbrack 4, 4 \rbrack ##

    I'm not sure where they are getting 9 linear combinations, but I'm pretty sure I'm misunderstanding the instructions of the problem.
     
  2. jcsd
  3. Jan 16, 2016 #2

    vela

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    Hint: ##3 \times 3 = 9##
     
  4. Jan 16, 2016 #3
    Okay, so the only thing I could think of using that hint would be that I can do other permutations of the c = 0, 1, 2 and d = 0, 1, 2?

    So maybe do 1 and 0, or 2 and 1, etc?
     
  5. Jan 16, 2016 #4

    Mark44

    Staff: Mentor

    Seems like it shoud be "with α = 0, 1, and 2, and β = 0, 1, and 2"

    BTW, it's much simpler to just use the [ and ] characters than typing lbrack and rbrack in LaTeX.
     
  6. Jan 16, 2016 #5

    Mark44

    Staff: Mentor

    Each choice of c (really ##\alpha##) can be paired with one of three possible d (really ##\beta##) values. How many combinations of the two does that make?
     
  7. Jan 16, 2016 #6
    Yeah, that's what I was getting at. I didn't know if the points could be interchanged, but I guess there isn't really any reason why they couldn't be.
     
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