Let $R:\mathbb R^3\to\mathbb R^3$ is reflection linear transformation on plane $\pi: x+2x+3z=0$:
a)Find a matrix of linear transformation $R$ using base $B=\{v1,v2,v3\}$ where $v_1=(1,1,-1), v_2=(-1,2,-1), v_3=(1,2,3)$
b)find a matrix of linear transformation using standard base of $R^3$
First I notice that $v_3$ is our normal vector so $v_1,v_2$ span the plane, and they orthogonal, then I know that $R(v_1)=v_1$, $R(v_2)=v_2$ and $R(v_3)=-v_3$, so I know how to do for that base, but I do not know how to do for standard base, I know for formula $Ref(v)=v-2\frac{v\cdot a}{a\cdot a}a$,here is just one vector a,but know I do not know is that formula for my case goes like this, if $qi=\frac{v_i}{||vi_||}$ ,$i=1,2$ now I put $Ref(e_1)=e_1-2\frac{e_1\cdot q_1}{q_1\cdot q_1}q_1-2\frac{e_1\cdot q_2}{q_2\cdot q_2}q2$, I put orthogonal vectors or orthonormal vectors?
Best Answer
Given the matrix $[R]_B^B$ and the basis $B$, you may derive the change of basis matrices from the standard basis $S$ to $B$ and back, i.e. you may determine the representation matrices $[id]_S^B$ and $[id]_B^S$.
The former is easily determined as
$$[id]_S^B=\begin{pmatrix}[v_1]_S &[v_2]_S &[v_3]_S\end{pmatrix}$$
where $[v_i]_S$ is the column vector of the coordinatization of $v_i$ with respect to the standard basis, i.e.
$$[id]_S^B=\begin{pmatrix}[v_1]_S &[v_2]_S &[v_3]_S\end{pmatrix}=\begin{pmatrix}1 &-1 &1\\1 &2 &2\\-1 &-1 &3\end{pmatrix}$$
For the latter, just note that $[id]_B^S={[id]_S^B}^{-1}$. Now, you have
$$[R]_S^S=[id]_S^B[R]_B^B[id]_B^S=[id]_S^B[R]_B^B{[id]_S^B}^{-1}$$