1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
olchik [2.2K]
3 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]3 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]

You might be interested in
Can someone please help me
yarga [219]

Answer:

A) <X

Step-by-step explanation:

X - it’s opposite the longest side

7 0
2 years ago
Use the diagram to copy and complete the statement. (#12, 14, 16)
Sergio [31]

11 is the biggest and the 22

8 0
3 years ago
What's the area of this triangle
Luda [366]

Answer:

D      A = (1/2) (y₂ - y₁)(x₃ - x₁)

Step-by-step explanation:

b = (y₂ - y₁)

h = (x₃ - x₁)

A = (1/2)bh

A = (1/2) (y₂ - y₁)(x₃ - x₁)

8 0
3 years ago
The Free Food Club holds weekly meetings. In chronological order, they've ordered 12, 9, 11, 10, 13, 8, 7, and 13 pizzas over th
Katena32 [7]

Answer:

10.375

Step-by-step explanation:

1.) Add up all the amount of pizzas |  12 + 9 + 11 +10+13+8+ 7,+ 13=83

2) Divide the total amount of pizzas by the amount of pizzas/amount of numbers of pizzas. | 83 divided by 8 =

10.375

6 0
3 years ago
Nancy also has a mirror that measures 20.32 centimeters long. What is the length of Nancy’s mirror in inches?
tino4ka555 [31]

Answer:

C: 8.00

Step-by-step explanation:

1 in = 2.54 cm

l = 20.32 × 1/2.54

l = 8.00 in

The length of Nancy’s mirror is 8.00 in.

3 0
3 years ago
Other questions:
  • Solve for x <br> 1/2(6x-4)=3x-2
    8·1 answer
  • Mrs. Hall went to work for the shirt factory on May 23, 1885. She earned nineteen cents per hour. If she worked thirteen hours p
    9·1 answer
  • Choose an equation that when paired with the equation below, will create a system of equations with infinitely many solutions, o
    9·1 answer
  • Which of the following deschbes the transformation from Figure 1 to Figure 2?
    12·1 answer
  • A bolt has 16-1/2 turns per inch. How many bolts would be in 2-1/2 inches?
    11·2 answers
  • (1 pt) Lucas cut a board that is ft long and ft wide. What is the area of the board? Express your answer in simplest form. A. sq
    12·1 answer
  • Let x be a variable, and let n be an arbitrary constant. What is the derivative of x^n?​
    9·1 answer
  • W= x/y-x solve for x
    14·1 answer
  • Suppose u and v are functions of x that are differentiable at x=0 and that u(0)= 7,u'(0)=-5,v(0)= -1, v'(0)= -4.
    7·1 answer
  • Find the slope of a line parallel to the line whose equation is 3Y- 5x =15
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!