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Homework Help: Some small linear algebra questions

  1. Apr 17, 2012 #1
    Hello, I just have some small questions about linear algebra that are giving me some slight grief.


    I know how to find the number of free variables and the basis of the solution space.

    But what is confusing me is the A transpose... When doing the problem, would I firstly take the transpose of A and then row reduce the matrix to find the free variables? or would I do it the other way around?



    I've solved part a)..

    But I am stuck on part b) I have no idea where to start the problem or what to actually use to actually attempt solving this part. So a suggestion as to what I can actually do, would be very helpful!

    thank you in advance.
  2. jcsd
  3. Apr 17, 2012 #2
    For the last part, one idea that comes to mind is that [itex]b_1 \in W[/itex] and [itex]b_2 \in W^{\perp}[/itex] means that [itex]b_1 = a \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} + b \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}[/itex] for some [itex]a,b \in \mathbb{R}[/itex] and similarly [itex]b_2[/itex] would be a linear combination of the basis you found for [itex]W^{\perp}[/itex].
  4. Apr 18, 2012 #3
    Well I think I've solved it, finally... If I'm correct, I got

    [itex]b = \begin{pmatrix} 2\\ 5 \\ 7 \\ 10 \end{pmatrix} = b_1 \begin{pmatrix} 8.1 \\ 10.2 \\ 12.3 \\ 14.4 \end{pmatrix} + b_2 \begin{pmatrix} -6.1 \\ -5.2 \\ -5.3 \\ -4.4 \end{pmatrix}[/itex]

    Also can anyone explain the first question to me? Thanks, in advance!
    Last edited: Apr 18, 2012
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