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| -rw-r--r-- | ClassJune26.page | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/ClassJune26.page b/ClassJune26.page index eea5dff..9bb4ded 100644 --- a/ClassJune26.page +++ b/ClassJune26.page @@ -77,14 +77,14 @@ b\end{array}\right)\\ 0\\ 1\end{array}\right)\\ = a\cdot1+b\cdot i\\ - = a+bi\end{eqnarray*} + = a+bi$ Put another way, we are using $1$ and $i$ as basis vectors. For -example, $ -\left(\begin{array}{c} +example, +$\left(\begin{array}{c} 3\\ 2\end{array}\right)=3(\rightarrow)+2(\uparrow)=3+2i$ -and $ -\overset{2}{\nwarrow}=\left(\begin{array}{c} +and +$\overset{2}{\nwarrow}=\left(\begin{array}{c} -\sqrt{2}\\ \sqrt{2}\end{array}\right)=-\sqrt{2}(\rightarrow)+\sqrt{2}(\uparrow)=-\sqrt{2}+\sqrt{2}i$ @@ -108,16 +108,16 @@ by some fixed complex number $\rho=r(\cos\theta+i\sin\theta)=a+bi$, which, as we saw before, is a dilation by $r$ plus a rotation by $\theta$. The distributive law says precisely that this is a linear transformation of the plane, viewed as a two-dimensional vector space. -And linear maps are given in rectangular coordinates by matrices:\[ -\left(\begin{array}{c} +And linear maps are given in rectangular coordinates by matrices: +$\left(\begin{array}{c} x\\ -y\end{array}\right)\rightsquigarrow\left(\begin{array}{cc} +y\end{array}\right)\mapsto\left(\begin{array}{cc} a & c\\ b & d\end{array}\right)\left(\begin{array}{c} x\\ y\end{array}\right)=\left(\begin{array}{c} ax+cy\\ -bx+dy\end{array}\right).\] +bx+dy\end{array}\right).$ What is the matrix corresponding to multiplication by $\rho$? Well, the first column is the image of $\left(\begin{array}{c} 1\\ |