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Math Help - Eigenspaces and eigenvectors

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    Eigenspaces and eigenvectors

    Find all 2 x 2 matrices A for which E(sub 7) = R^2.

    Does E(sub 7) mean there are 7 eigenspaces? Can someone clarify this problem further?
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    Quote Originally Posted by noles2188 View Post
    Find all 2 x 2 matrices A for which E(sub 7) = R^2.

    Does E(sub 7) mean there are 7 eigenspaces? Can someone clarify this problem further?

    I think it'd be a good idea YOU first find out what is E_7 since it is not, apparently, standard notation. Ask you book/teacher.

    Tonio
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    According to my book:

    E(sub 7) = ker(A - 7I(sub 2)) = {v[IMG]file:///C:/Users/Geoffrey/AppData/Local/Temp/moz-screenshot.png[/IMG]in R^2 : Av = 7v}

    7 is an eigenvalue and, to me, the question is asking to find matrices A such that Av = 7v. How would I go about this?
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    E_7 denotes the eigenspace associated with the eigenvalue 7.
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    Quote Originally Posted by noles2188 View Post
    According to my book:

    E(sub 7) = ker(A - 7I(sub 2)) = {v[IMG]file:///C:/Users/Geoffrey/AppData/Local/Temp/moz-screenshot.png[/IMG]in R^2 : Av = 7v}

    7 is an eigenvalue and, to me, the question is asking to find matrices A such that Av = 7v. How would I go about this?

    Ok then, let A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right  ) be such a matrix, then \left(\begin{array}{cc}a&b\\c&d\end{array}\right)\  left(\begin{array}{c}x\\y\end{array}\right)=\left(  \begin{array}{c}7x\\7y\end{array}\right)\,\,\foral  l\,x\,,\,y\in \mathbb{R} , since they want E_7=\mathbb{R}^2.

    Well, now choose wisely x\,,\,y\in\mathbb{R} to find the coefficients of the matrix. For example, choosing x=1\,,\,y=0 gives us at once that a=7\,,\,c=0...

    Tonio
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