Lauren1234 0 Posted January 20, 2020 Share Posted January 20, 2020 How does one show the below matrix is a linear transformations I know I need to multiply something by (0,1) and (1,0) Link to post Share on other sites

timo 583 Posted January 22, 2020 Share Posted January 22, 2020 An operator f(...) is linear if f(A+B) = f(A) + f(B) and f(a*A) = a*f(A), with addition and multiplication being the addition of two vectors and their multiplication with a real number, respectively, in your case. Alternate form of the same statements for a matrix M, vectors x, y, and a scalar a: M(x+y) = Mx + My, M(a*x) = a*(Mx). When interpreted as an operator V -> V, matrices are always linear. But it should be straightforward to explicitly show that for your given matrix by starting from one side of the two defining equations and rearranging until you get the other side. Link to post Share on other sites

Country Boy 70 Posted February 10, 2020 Share Posted February 10, 2020 (edited) Yes, the "natural basis" for [tex]R^2[/tex] is {(1, 0), (0, 1)}. Rotating (1, 0) through $\pi/3$ radians counter-clockwise gives [math](cos(\pi/3), sin(\pi/3))= (1/2, \sqrt{3}/2)[/math] and rotating (0, 1) through $\pi/3$ radians counter clockwise gives $(cos(4\pi/3), sin(4\pi/3)= (-sin(\pi/3), cos(\pi/3))= (-\sqrt{3}/2, 1/2)$. To represent that as a matrix, you need $\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ so that $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix}1 \\ 0 \end{pmatrix}= \begin{pmatrix}a \\ c \end{pmatrix}= \begin{pmatrix}1/2 \\ \sqrt{3}/2}\end{pmatrix}$ so a= 1/2 and $c= \sqrt{3}/2$. And, similarly $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix}0 \\ 1 \end{pmatrix}= \begin{pmatrix}b \\ c \end{pmatrix}= \begin{pmatrix}\sqrt{3}/2, 1/2\end{pmatrix}$ so $b= -sqrt{3}/2$ and $d= 1/2$. Edited February 10, 2020 by Country Boy Link to post Share on other sites

joigus 461 Posted May 22, 2020 Share Posted May 22, 2020 In other words, ask yourself, is it true that, \[\left[R_{\pi/3}\right]_{\mathcal{B}}\left(\lambda\boldsymbol{u}+\mu\boldsymbol{v}\right)=\lambda\left[R_{\pi/3}\right]_{\mathcal{B}}\boldsymbol{u}+\mu\left[R_{\pi/3}\right]_{\mathcal{B}}\boldsymbol{v}\] for any u, v, lambda and mu? Link to post Share on other sites

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