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object by 60° about the origin and abount point Aa Find the matrix that represents rotation...
4. Find the 3 x 3 matrix that produces a rotation by 60° about the point...
4. Find the 3 x 3 matrix that produces a rotation by 60° about the point (-4,-6) using homogeneous coordinates. (You do not have to multiply the matrices).
6a Perform a 60° rotation of point p(3, 5) (a) about the origin and (b) about P(2,2). b. Find the normalization tra...
6a Perform a 60° rotation of point p(3, 5) (a) about the origin and (b) about P(2,2). b. Find the normalization transfgation that maps a and upper right corner is at (4,6) onto (a) a viewport that is the entire normalized device screen and (b) a viewport that has lower left corner at (0,0) and upper right corner (1/2,1/2). Clipping against rectangular windows whose sides are aligned with the x andy axes involves computing intersections with vertical and horizontal lines....
Let T : R2 → R2 be the linear transformation given by T(v) = Av that consists of a counterclockwise rotation about the origin through an angle of 30 2, Find the matrix that produces a countercloc...
Let T : R2 → R2 be the linear transformation given by T(v) = Av that consists of a counterclockwise rotation about the origin through an angle of 30 2, Find the matrix that produces a counterclockwise rotation about the origin through an angle of 30°. Be sure to give the EXACT value of each entry in A. a. b. Plot the parallelogram whose vertices are given by the points A(0, 0), B(4, 0), C(5, 3), and D 1, 3)...
3. In-Class Exercises (75 minutes (a) Find the rotation matrix that rotates the vector A into the...
3. In-Class Exercises (75 minutes (a) Find the rotation matrix that rotates the vector A into the vector B, where, in the home repre- sentation, 134 ー4 (b) Find the inverse of the rotation matrix c) What happens to the scalar product of A and B under this rotation?
3. In-Class Exercises (75 minutes (a) Find the rotation matrix that rotates the vector A into the vector B, where, in the home repre- sentation, 134 ー4 (b) Find the inverse...
1 point) Using homogeneous coordinates, the matrix A which rotates the point (-4,-5,-1) about the...
1 point) Using homogeneous coordinates, the matrix A which rotates the point (-4,-5,-1) about the z-axis through an angle of counterclockwise as viewed fronm the positive z-axis and then shifts the result by-5 in the x-direction, 4 in the y-direction, and -2 in the z-direction, is given by The image of (-4,-5,-1) under this transformation is 0 , and Z =
1 point) Using homogeneous coordinates, the matrix A which rotates the point (-4,-5,-1) about the z-axis through an angle...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix whi...
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents ф with respect to the standard basis. (b) what the the eigenvalues and eigenvectors of67 (c) If we consider A to be a complex matrix (since all real numbers are complex numbers), what are the eigenvalues and eigenvectors of A?
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents...
20. Find the coordinates of the vertices of the figure below after a rotation 270 degrees...
20. Find the coordinates of the vertices of the figure below after a rotation 270 degrees clockwise about the origin. A) (1, 2). (2,4), K(4.3), 2(4,1) B) 10.-1), 14-1, -3), 6(-3,-2), L(-3,0) C)/(-1,0). J{ 3,1),(-2,3), (0, 3) D) /1,3), ((-1.2.2-1,0), 7(2.1) Mark your answer to #20 here. OA OB
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin,...
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...
(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the a...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
4. Find the 3 x 3 matrix that produces a rotation by 60° about the point (-4,-6) using homogeneous coordinates. (You do not have to multiply the matrices).
6a Perform a 60° rotation of point p(3, 5) (a) about the origin and (b) about P(2,2). b. Find the normalization transfgation that maps a and upper right corner is at (4,6) onto (a) a viewport that is the entire normalized device screen and (b) a viewport that has lower left corner at (0,0) and upper right corner (1/2,1/2). Clipping against rectangular windows whose sides are aligned with the x andy axes involves computing intersections with vertical and horizontal lines....
Let T : R2 → R2 be the linear transformation given by T(v) = Av that consists of a counterclockwise rotation about the origin through an angle of 30 2, Find the matrix that produces a counterclockwise rotation about the origin through an angle of 30°. Be sure to give the EXACT value of each entry in A. a. b. Plot the parallelogram whose vertices are given by the points A(0, 0), B(4, 0), C(5, 3), and D 1, 3)...
3. In-Class Exercises (75 minutes (a) Find the rotation matrix that rotates the vector A into the vector B, where, in the home repre- sentation, 134 ー4 (b) Find the inverse of the rotation matrix c) What happens to the scalar product of A and B under this rotation?
3. In-Class Exercises (75 minutes (a) Find the rotation matrix that rotates the vector A into the vector B, where, in the home repre- sentation, 134 ー4 (b) Find the inverse...
1 point) Using homogeneous coordinates, the matrix A which rotates the point (-4,-5,-1) about the z-axis through an angle of counterclockwise as viewed fronm the positive z-axis and then shifts the result by-5 in the x-direction, 4 in the y-direction, and -2 in the z-direction, is given by The image of (-4,-5,-1) under this transformation is 0 , and Z =
1 point) Using homogeneous coordinates, the matrix A which rotates the point (-4,-5,-1) about the z-axis through an angle...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents ф with respect to the standard basis. (b) what the the eigenvalues and eigenvectors of67 (c) If we consider A to be a complex matrix (since all real numbers are complex numbers), what are the eigenvalues and eigenvectors of A?
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents...
20. Find the coordinates of the vertices of the figure below after a rotation 270 degrees clockwise about the origin. A) (1, 2). (2,4), K(4.3), 2(4,1) B) 10.-1), 14-1, -3), 6(-3,-2), L(-3,0) C)/(-1,0). J{ 3,1),(-2,3), (0, 3) D) /1,3), ((-1.2.2-1,0), 7(2.1) Mark your answer to #20 here. OA OB
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
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