Find the matrix A of the linear transformation T from R² to R² that rotates any vector through an angle of 45° in the clockwise direction. A

Find the matrix A of the linear transformation T from R? to R? that rotates any vector through an angle of 120° in the clockwise direction. A:

Let T: R² R² be the linear transformation that first rotates points clockwise through 120° (2/3 radians) and then reflects points through the line y=z. Find the standard matrix A for T. A →

Let's create a matrix that reflects around a line that is perpendicular to the vector n = [0.375, 0.927] in 2D, this line passes though the origin. Now that you have both rows, what is the image of the point (1, 2) under this transformation? Enter the x-coordinate, round it to 3 decimal places.

Find the image of the point (x, y) under the matrix transformation that first reflects the point with respect to the x-axis, then rotates the resulting point about the origin through an angle of T/3, and then reflects the resulting point with respect to the line y = x.

Assume that T is a linear transformation. Find the standard matrix of T. П T: R² R², rotates points (about the origin) through radians. 2 A= (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)

Let T be the linear transformation which takes a vector in R² and does the following in sequence: • Shears it by a factor of 3 in the x-direction, • Reflects it over the y-axis, • Rotates it 90° clockwise about the origin, and finally • Dilates it by a factor of 9. a) Find the matrix associated with T. b) Prove that the transformation is linear through explicit calculation.

Please solve all three parts

Let T : R? → R? be the linear transformation defined by a counter clockwise rotation of 30° in R?. (a) Find the standard matrix A for the linear transformation. (b) Use A to find the image of the vector v = (c) Sketch the graph of v and its image. (1, 2).

Let T be the linear transformation associated with the matrix: 0 1 -1 0 Find T(V) if V is the vector: 2 2 Is the transformation a rotation or a reflection? Justify why.