Respuesta :
The matrix for the combined rotation transformation of 45° and dilation
of [tex]\sqrt{2}[/tex] gives the matrix for the transformation as follows;
[tex]Matrix \ for \ the \ rotation \ and \ dilation \ is;\left[\begin{array}{cc}1 &-1\\1&1\end{array}\right][/tex]
How can the transformation matrix be found?
The angle by which the transformation rotates about the origin = 45°
The dilation following the rotation = A factor of √2
Required:
The matrix for the transformation
Solution:
The dilation matrix for a scale factor of [tex]\mathbf{\sqrt{2}}[/tex] is presented as follows;
[tex]\mathbf{\left[\begin{array}{cc}\sqrt{2} &0\\0&\sqrt{2} \end{array}\right]}[/tex]
[tex]The \ transformation \ for \ a \ rotation \ of \ 45^{\circ} \ is; \left[\begin{array}{cc}cos(45^{\circ})&-sin(45^{\circ})\\sin(45^{\circ})&cos(45^{\circ})\end{array}\right][/tex]
[tex]\left[\begin{array}{cc}cos(45^{\circ})&-sin(45^{\circ})\\&\\sin(45^{\circ})&cos(45^{\circ})\end{array}\right] = \mathbf{\left[\begin{array}{cc}\dfrac{\sqrt{2} }{2} &-\dfrac{\sqrt{2} }{2} \\&\\ \dfrac{\sqrt{2} }{2} &\dfrac{\sqrt{2} }{2} \end{array}\right]}[/tex]
Which gives;
[tex]\left[\begin{array}{cc}\dfrac{\sqrt{2} }{2} &-\dfrac{\sqrt{2} }{2} \\&\\ \dfrac{\sqrt{2} }{2} &\dfrac{\sqrt{2} }{2} \end{array}\right] \times \left[\begin{array}{cc}\sqrt{2} &0\\0&\sqrt{2} \end{array}\right] = \left[\begin{array}{cc}\dfrac{\sqrt{2} \times \sqrt{2} }{2} &-\dfrac{\sqrt{2} \times \sqrt{2} }{2} \\&\\ \dfrac{\sqrt{2} \times \sqrt{2} }{2} &\dfrac{\sqrt{2} \times \sqrt{2} }{2} \end{array}\right] = \left[\begin{array}{cc}1 &-1\\1&1\end{array}\right][/tex]
The required matrix for the transformation is therefore;
- [tex]\left[\begin{array}{cc}1 &-1\\1&1\end{array}\right][/tex]
Learn more about other forms of transformation matrices here:
https://brainly.com/question/10212935
