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Frustrated with finding basis

  1. Dec 8, 2011 #1
    Hey guys

    There are so many of these damn "Find a basis" questions and I can't get any of them because we never directly learned how...or she never showed us in class...my final exam is tomorrow.

    Here are some examples of questions:

    http://184.154.165.18/~devilthe/uploads/1323453294.png

    http://184.154.165.18/~devilthe/uploads/1323430492.png

    Part D

    I have zero idea how they solve these...

    I know that a basis is a linearly independent spanning set...but how to solve these questions? No idea.

    Can there be more than one basis for a question?

    Thanks,
    Elliott
     
  2. jcsd
  3. Dec 8, 2011 #2

    Mark44

    Staff: Mentor

    A basis is a set of vectors (or matrices in this problem) that is linearly independent and spans the (sub)space. A vector (sub)space can have many bases.

    For d, start by playing with the equation that defines the subspace V, using an arbitrary matrix X, where
    [tex]X = \begin{bmatrix}a & b \\ c & d \end{bmatrix}[/tex]
     
  4. Dec 9, 2011 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Any polynomial, of degree 2, is of the form [itex]p(x)= ax^2+ bx+ c[/itex]. [itex]p'(x)= 2ax+ b[/itex] Requiring that p'(1)= 0 means that 2a+ b= 0 so b= -2a. That is, for any p in this set, [itex]p(x)= ax^2- 2ax+ c= a(x^2- 2)+ c(1)[/itex]. Now, what is a basis for that set?

    Any 2 by 2 matrix is of the form
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]

    We require that
    [tex]\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}[/tex]

    Multiply those, set corresponding term equal, and see what you get.
     
  5. Dec 9, 2011 #4
    Thanks guys.

    Anyway I just did my linear exam today so hopefully I never have to see linear ever again!
     
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