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Question

# A transformation T: $V_n + V_n$ is described as, T rotates every point through the same angle $\phi$ about the origin. That is, T maps a point with polar coordinates (r, 0) onto the point with polar coordinates $(r, \theta+\phi)$, where $\phi$ is fixed. Also, Tmaps 0 onto itself. Determine whether T is linear. If T is linear, describe its null space and range, and compute its nullity and rank.

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Let $(r,\theta)$ be a point in $V_2$, in polar coordinates. Then, its Cartesian coordinates are $(x,y)$, where

\begin{align*}x&=r\cos\theta,\\ y&=r\sin\theta.\end{align*}

The function $T$ is defined such that, in polar coordinates, $T(r,\theta)=(r,\theta+\phi)$, where $\phi$ is constant. In Cartesian coordinates, this means that $T(x,y)=(x_0,y_0)$, where

\begin{align*}x_0&=r\cos(\theta+\phi)\\ &=r(\cos\theta\cos\phi-\sin\theta\sin\phi)\\ &=\cos\phi\cdot r\cos\theta-\sin\phi\cdot r\sin\theta\\ &=\cos\phi\cdot x-\sin\phi\cdot y,\\ y_0&=r\sin(\theta+\phi)\\ &=r(\sin\theta\cos\phi+\cos\theta\sin\phi)\\ &=\cos\phi\cdot r\sin\theta+\sin\phi\cdot r\cos\theta\\ &=\sin\phi\cdot x+\cos\phi\cdot y.\end{align*}

Therefore, in Cartesian coordinates, $T(x,y)=(\cos\phi\cdot x-\sin\phi\cdot y,\sin\phi\cdot x+\cos\phi\cdot y)$. We know that any function $T:V_2\to V_2$ of the form $T(x,y)=(ax+by,cx+dy)$ must be linear, so $T$ is linear.

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