Find the matrix of $T$ with respect to standard basis of $V$.

Question

Let $T:V\to V$ be the rotation by an angle $\theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=\Bbb R^3$

Find the matrix of $T$ with respect to standard basis of $V$.

I know that the equation of the plane passing through the origin perpendicular to $(1,2,3)$ is $x+2y+3z=0$ but I dont know how to find the matrix .

I really dont understand where does $T$ map the vector $(1,0,0)$ .

Can someone kindly help me?

Hints will suffice.I dont see any way out

Answer 1

Extend $v_1=\frac{1}{\sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.

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The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $\|v_1\|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:

1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization
2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1\times v_2$.

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Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.

Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $\theta$ in this plane and we have \begin{align*} T(v_2) &= \cos(\theta)\, v_2 + \sin(\theta)\, v_3, \\ T(v_2) &= -\sin(\theta)\, v_2 + \cos(\theta)\, v_3. \end{align*} Thus, the matrix of $T$ with respect to the basis $B$ is given by $$M(T)_B = \begin{bmatrix} 1 & 0&0\\0 &\cos \theta&-\sin \theta\\0& \sin\theta &\cos\theta\end{bmatrix}.$$ The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 \,|\, v_2 \,|\, v_3)$ with columns $v_1, v_2, v_3$.

Hence, the matrix of $T$ with respect to the standard basis is $$ M(T)_S = C_{S,B} \,M(T)_B \,C_{B,S} = C_{S,B} \,M(T)_B \,C_{S,B}^{-1}. $$