Question

Describe the images and kernels of the transformations in Exercisesthrough geometrically.

25. Rotation through an angle$\frac{\mathbf{\pi }}{\mathbf{4}}$ of in the counterclockwise direction (in${\mathbf{ℝ}}^{\mathbf{2}}{\mathbf{ℝ}}^{\mathbf{2}}$).

Short answer

The kernel is $\left\{\overrightarrow{0}\right\}$, image is all of ${\mathrm{ℝ}}^2$.

Step-by-step solution

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If fis a function from X to Y, then

$$\begin{gathered} image\left(f\right)=\left\{f\left(x\right):x\mathrm{in}X\right\} \\ =\left\{binY:b=f\left(x\right),\mathrm{for}\mathrm{some}x\mathrm{in}X\right\} \end{gathered}$$

The Kernel of the transformation results in $\left\{\overrightarrow{0}\right\}$, as the reflection is its own inverse.

As the inverse of the reflection exists thus, the image will be ${\mathrm{ℝ}}^2{\mathrm{ℝ}}^2{\mathrm{ℝ}}^2$. The kernel is of dimension $\left\{\overrightarrow{0}\right\}$ so, the image will be of ${\mathrm{ℝ}}^2$.

Final answer.

The kernel is $\left\{\overrightarrow{0}\right\}$, image is all of ${\mathrm{ℝ}}^2{\mathrm{ℝ}}^2{\mathrm{ℝ}}^2$.