Find the matrices of the following linear transformations from R³ to R³. The orthogonal projection onto the y-axis: 393
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- 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 = 9-Let T: R2-R2 be a linear transformation that first reflects vectors about the y-axis and then rotates vectors TT/2 radians counterclockwise. Find the standard matrix for TFind the matrices of the following linear transformations from R³ to R³. The orthogonal projection onto the y-axis: The reflection across the xz-plane:
- Find the standard matrix for the linear transformation T: R → R´ that rotates points about the origin by 2 radians. Check AnswerFind the metrix of given linear transformationWrite down the 2 × 2 matrices that represent the following linear transformations ofthe plane. Also draw the image of the (first quadrant) unit square 1 under each transformation.(a) A dilation with horizontal dilation factor 2 and vertical dilation factor 1/2.(b) A vertical shear with factor −1/2. (c) A rotation of 450 anticlockwise. (d) A reflection across the line at angle −300 to the X-axis. (e) The transformation T(x, y) = (2x + 6y, x + 3y).
- Map the straight line joining A (-2 + j3) and B (1 + j2) in the z-plane onto the w-plane using the transformation w= (-3 +j)z+ (2 + j4). State the magnification, rotation, and translation.For the linear transformation С represented by the equation x - 2y = 0, find a vector v parallel to e, and a vector w perpendicular to C. Make your vectors v and w simple so that your computations are not overly complicated.Find the Jacobian J of the transformation
- Let T : R → R´ be the linear transformation that first rotates points clockwise through 45° (t/4 radians) and then reflects points through the line y = x. Find the standard matrix A for T. -sqrt2/2 -sqrt2/2 A = sqrt2/2 sqrt2/2Let T:R? → R² be the linear transformation that changes the length of a vector to one-third of its original length, and reflects the vector across the y-axis. Find the standard matrix for T.Use a rectangular coordinate system to plot u = and their images under the given transformation T. Describe geometrically what T does to each vector x in R2. 0 0 X1 T(x) = 0 1 X2 Which graph below shows u and its image under the given transformation? O A. О в. Oc. OD. AX2 Which graph below shows v and its image under the given transformation? O A. О в. Oc. OD.