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Linear Systems: Linear combinations

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine if b is a linear combination of a1, a2, and a3.

    a1= [1,-2, 0]
    a2 = [0, 1, 2]
    a3 = [5, -6 8]

    b= [2, -1, 6]


    3. The attempt at a solution

    Alright, well I used an augmented matrix to solve the problem, and after reducing it completely, the matrix looked like this:

    | 1 0 5 |2|
    | 0 1 4 |3|
    | 0 0 0 |0|

    Solving for the c's, I got:
    (Where the number to the right of the c denote which one instead of a multiplication number)
    c1+ 5c3=2
    c2+4c3=3
    0c3=0

    I found that b is a combination of a1 and a2 if c3 is set to 0, however, since c3 is a free variable, and therefore can be anything, then b wouldn't be a lin. combination if c3 be anything other than 0.

    I'm having a hard time understanding this. Is it not a linear combination because there are other possible answers that wouldn't make it so? Or is it a linear combination because atleast one possibility works.

    Thank you for any help.
     
  2. jcsd
  3. Sep 20, 2012 #2

    Mark44

    Staff: Mentor

    No, this isn't true. c3 can be set to any value. Of course, this will give different values for c1 and c2. The point is that there are lots of linear combinations of a1, a2, and a3 that produce b.

    One thing to notice is that if c3 = 0, then c1 = 2 and c2 = 3. What this says is that b = 2a1 + 3a2.

    Geometrically, b lies in the same plane as a1 and a2.
     
  4. Sep 20, 2012 #3
    Ah! I understand.

    Thanks, I forgot to also change the values for c1 and c2 when I picked another free c3.

    What had me confused is that the book says it's not a combination. The answer on the back clearly states it in bold. However,the book is probably wrong (Which now a-days I find they often are in regards to answers in the back)
     
  5. Sep 20, 2012 #4

    Mark44

    Staff: Mentor

    First, make sure you are working the right problem.
    Second, if you're working the same problem as in the book, check your answers. In this case, set c3 to some value (0 is convenient here) and verify that 2a1 + 3a2 actually adds up to b.

    Try a different linear combination, by setting c3 to some other value (and calculating values for the two other constants). See if that linear combination of a1, a2, and a3 also ends up with b.
     
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