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?
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 $\displaystyle A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right )$ be such a matrix, then $\displaystyle \left(\begin{array}{cc}a&b\\c&d\end{array}\right)\ left(\begin{array}{c}x\\y\end{array}\right)=\left( \begin{array}{c}7x\\7y\end{array}\right)\,\,\foral l\,x\,,\,y\in \mathbb{R}$ , since they want $\displaystyle E_7=\mathbb{R}^2$.
Well, now choose wisely $\displaystyle x\,,\,y\in\mathbb{R}$ to find the coefficients of the matrix. For example, choosing $\displaystyle x=1\,,\,y=0$ gives us at once that $\displaystyle a=7\,,\,c=0$...
Tonio