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olchik [2.2K]
2 years ago
10

Let T:ℝ2→ℝ2 be the linear transformation that first rotates points clockwise through 45∘ (????/4 radians) and then reflects poin

ts through the line y=x. Find the standard matrix ???? for T.
Mathematics
1 answer:
Alisiya [41]2 years ago
3 0

Answer:

T = \left[\begin{array}{ccc}-\frac{1}{\sqrt{2} } &\frac{1}{\sqrt{2} }\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\end{array}\right]

Step-by-step explanation:

Let General Transformation matrix be denoted as T

Step 1: Clockwise rotation of 45 degrees

General counterclockwise rotation matrix in 2-dimension is given as

                                        R(\theta)=\left[\begin{array}{ccc}cos\theta & - sin\theta\\sin\theta&cos\theta\\\end{array}\right]

For clockwise rotation we need to insert θ as negative in the above matrix. Therefore, the resulting matrix is

                                        R(-\theta)=\left[\begin{array}{ccc}cos\theta & sin\theta\\-sin\theta&cos\theta\\\end{array}\right]

as sin(-θ) = -sin (θ) and cos(-θ) = cos (θ)

For 45 degrees

sin(45)  = \frac{1}{\sqrt{2} }   and   cos(45)  = \frac{1}{\sqrt{2} }

                                       R(-45)=\left[\begin{array}{ccc}\frac{1}{\sqrt{2} }  & \frac{1}{\sqrt{2} }\\-\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\\\end{array}\right]

Step 2: Reflection through line y = x

This type of reflection maps (x,y)→(y,x)

Therefore the general matrix is

                                           R(x,y)=\left[\begin{array}{ccc}0&1\\1&0\end{array}\right]

Step 3: General Transformation Matrix

T = R(x,y) R(-θ)

                                    T=\left[\begin{array}{ccc}0&1\\1&0\end{array}\right] \left[\begin{array}{ccc}\frac{1}{\sqrt{2} }  & \frac{1}{\sqrt{2} }\\-\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\\\end{array}\right]

                                           T = \left[\begin{array}{ccc}-\frac{1}{\sqrt{2} } &\frac{1}{\sqrt{2} }\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\end{array}\right]

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