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Thread: Basis and Dimension of Vector Spaces

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    Member Maccaman's Avatar
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    Basis and Dimension of Vector Spaces

    Give a basis and the dimension of each of the following vector spaces...

    (a) The space of 3 3 matrices which are invarient under a 90-degree clockwise rotation; that is, the matrices satisfying
    $\displaystyle \begin{pmatrix} a&b&c\\d&e&f\\g&h&i\end{pmatrix} = \begin{pmatrix} g&d&a\\h&e&b\\i&f&c\end{pmatrix} $

    (b) The space of polynomials in $\displaystyle P_n(\mathbb{R}) (n\geq 2) $ which are divisible by $\displaystyle x^2 +1 $. (i.e. they can be written as the product of $\displaystyle x^2 + 1 $ with another polynomial).
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    Quote Originally Posted by Maccaman View Post
    Give a basis and the dimension of each of the following vector spaces...

    (a) The space of 3 3 matrices which are invarient under a 90-degree clockwise rotation; that is, the matrices satisfying
    $\displaystyle \begin{pmatrix} a&b&c\\d&e&f\\g&h&i\end{pmatrix} = \begin{pmatrix} g&d&a\\h&e&b\\i&f&c\end{pmatrix} $
    so we have $\displaystyle a=c=g=i$ and $\displaystyle b=d=f=h.$ so an elment of your vector space is in the form: $\displaystyle aX + bY + eZ,$ where:

    $\displaystyle X=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix}, \ \ Y=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},$ and $\displaystyle Z=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}.$ so the dimension is 3 and a basis for the space is $\displaystyle \{X,Y,Z \}.$


    (b) The space of polynomials in $\displaystyle P_n(\mathbb{R}) (n\geq 2) $ which are divisible by $\displaystyle x^2 +1 $. (i.e. they can be written as the product of $\displaystyle x^2 + 1 $ with another polynomial).
    an element of of your vector space here is in the form $\displaystyle (x^2+1)(c_{n-2}x^{n-2} + \cdots + c_1 x + c_0)= c_{n-2}(x^n + x^{n-2}) + \cdots + c_1(x^3+x) + c_0(x^2+1).$ it's clear now that the

    dimension of your space is $\displaystyle n-1$ and a basis is $\displaystyle \{x^n + x^{n-2}, \cdots , x^3 + x , x^2 + 1 \}.$
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