Problem #4: For the linear transformatation T: R³ → R³, the images of the following three vectors are given: T( 888004 -4 Find the standard matrix for the transformation.
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- Let TA: 23 be the matrix transformation corresponding to A=[311124]. Find TA(u) and TA(v), where u=[12] v=[32].Write 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).1) Derive the matrix that performs the following consecutive transformations(in augmented form):1- Translation of 2 units along x-axis.2- Rotation by 90° about z-axis.3- Rotation by 90° about y-axis.
- In this problem, allow T1: ℝ2→ℝ2 and T2: ℝ2→ℝ2 be linear transformations. Find Ker(T1), Ker(T2), Ker(T3) of the respective matrices:Suppose T: R² R² is a linear transformation that first reflects points across the ₁-axis and then reflects points across the line x2 = -₁. Find the standard matrix, A, of T. [+] AWrite 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 reflection across the line at angle −300 to the X-axis. (b) The transformation T(x, y) = (2x + 6y, x + 3y).
- From the cubic polynomials, P3 to the fourth-degree polynomials P4, what matrix represents multiplication by 2 + 3t? The columns of the 5 by 4 matrix A come from applying the transformation to 1, t, t², t³.Find the matrix of a linear transformation that would take the line 2x -5y = 0 to 7x -3y = 0You need to rotate point bP1=n [10 10 0] 60 degrees on the X axis and 10 units on the Z axis. Write the transformation matrix and graph of the resulting vector
- Find the standard matrix of each of the following linear transformations. (a) T: R² R², which reflects each point in R² across the line x₁ = 0 (i.e. the 22-axis) - -> (b) S: R² R², which rotates each point in R² clockwise by 90° and then dilates it by a factor of 7 (c) P: R³ R³, which projects each point in R³ onto the plane z₁ = 0 (i.c. the 2x3-plane)Answer the following two parts. (a) Find the matrix of the linear transformation, b) Find the matrix of linear transformation that rotates a vector in R2 clockwise by π 3 .4. Do the following transformations on the rectangle shown in figure below in sequence. a. Shear the rectangle about y axis by shy =2. Using Shear Matrix and Equation b. After Change the position of the rectangle by a vector(2,3). c. Perform operation i) and i) for the rectangle by finding its Composite Matrix. Y axis 6. (1,4) (4,4) 4 (1,2) (4,2) Хаxis 1 4 5. An object is in location (1,3), (1,6), (3,3) and (3,6). Do two repeated translation operations on the object with the scaling factors, (2,1) and (4,3). Write the new positions. 6. An object is in location (0,0), (1,0), (0,1) and (1,1). Write the new positions after shearing the object about x-direction. Take shear parameter along x-direction shx=4.Then apply reflection about the line x=0(the Y axis). 7. Write tests performed on the vertices of polygon edges by Sutherland Hodgeman Polygon Clipping algorithm. Apply Left, Right, Bottom and Top boundary clipping in the following polygon as is done by Sutherland Hodgeman Polygon…
