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Expressing a vector as a linear combinatino of another 2 vectors, how did i miss this

  1. Nov 26, 2005 #1
    Okay this problem seems easy, and i t hought i understood how to do it, but no.

    Express the vector v= [14 -12]^T as a linear combination of:
    x = [2 -1]^T; y = [-1 3];
    v = _x + _y;
    _ means thats were they want me to put an answer.

    So i row reduced
    2 -1
    -1 3
    which is
    1 0
    0 1

    so why wouldn't the answer just be
    v = 14x -12y

    becuase
    14 * [1 0] = [14 0];
    -12*[0 1] = [0 -12];
    which is [14 -12] if u add them so what the?
     
  2. jcsd
  3. Nov 26, 2005 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Because x and y are NOT [1, 0] and [0,1]! Saying that the "row-reduce" to [1,0] and [0,1] does not mean they are equal to them.

    I would suggest not row reducing at all:
    [tex]\alpha x+ \beta y= [2\alpha, -\alpha]+ [-\beta,3\beta][/tex]
    [tex]= [2\alpha- \beta,-\alpha+ 3\beta]= [14, -12][/tex]
    so you must have [itex]2\alpha- \beta= 14[/itex] and [itex]-\alpha+ 3\beta= -12[/itex]. Can you solve those two equations?
     
  4. Nov 26, 2005 #3
    Ahhh, i c, thank you Ivy!! It worked out great!
     
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