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Finding the basis for a vector space

  1. Aug 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Find a basis for the following vector space:

    The set of 2x2 matrices A such that CA=0 where C is the matrix : 1 2
    3 6


    3. The attempt at a solution

    I multiplied C by a general 2x2 matrix : a b and got 4 equations but two of these equations are the
    c d


    same and it seems as I am going around in circles. I know I need to find a set that is linearly independent and spans the original set but I'm not sure how to proceed.

    Help would be greatly appreciated :)

    Cheers
     
  2. jcsd
  3. Aug 20, 2013 #2

    CompuChip

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    Homework Helper

    That sounds exactly like what you want. Clearly not any 2 x 2 matrix will do, and indeed the two equations you have will fix two relations between a, b, c and d. But since there is not a single unique solution, you should also have at least one parameter which you can vary.

    Talking with a practical example is probably clearer, so can you show us which equations you got?

    Have a matrix by the way (quote my post to see the code): ##\begin{pmatrix} a & b \\ c & d \end{pmatrix}##
     
  4. Aug 20, 2013 #3
    sorry had no idea how to matrices on the forum

    C = ##\begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}##

    A(my general matrix) = ##\begin{pmatrix} a & b \\ c & d \end{pmatrix}##

    CA=0 so when I multiplied, I got

    a+2c=0
    b+2d=0
    3a+6c=0
    3b+6d=0

    But two of those equations are the same and I don't know what to do after that.
     
  5. Aug 20, 2013 #4

    CompuChip

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    Right, so if you get all the information you can from those equations, you can write your matrix A in terms of two variables only - for example, just a and b.

    What does A look like then?
     
  6. Aug 20, 2013 #5
    Makes sense now - got it!

    Thanks a lot - help was much appreciated :)
     
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