Find the matrix for the linear transformation which reflects every 2-dimensional vector across the y axis...
A 2 x 2 matrix A is associated with the transformation that reflects vectors in the plane across the line y = 2x. Sketch the vector field (use your discretion and draw vectors until you believe that you have captured the big picture) corresponding to A. In other words, for every ~w, we place A ~w at the head of ~w.
Problem 3. Construct a single 2 x 2 matrix which defines the transformation on R?, and find the image of the point C) under the transformation. a. A transformation which moves points to one quarter of their original distance to the origin. b. A transformation which first rotates points counterclockwise through an angle 31/2, then reflects them across the y-axis. A transformation which first reflects points across the y-axis, then rotates them counterclockwise through an angle 31/2.
Let T:R2 → R2 be the linear transformation that first reflects points through the x-axis and then then reflects points through the line y = -x. Find the standard matrix A for T.
(12) (after 3.3) (a) Find a linear transformation T. Rº Rº such that T(x) = Ax that reflects a vector (1), 12) about the Tz-axis. (b) Find a linear transformation SR2 R2 such that T(x) = Bx that rotates a vector (2, 2) counterclockwise by 135 degrees. (c) Find a linear transformation (with domain and codomain) that has the effect of first reflecting as in (a) and then rotating as in (b). Give the matrix of this transformation explicitly. How...
Let t be the linear transformation t: r2 -> r2 that reflects a vector about the line y=x. Find the eigenvalue and eigenvectors of T. How can you interpret this geometrically?
(Note: Each problem is worth 10 points). 1. Find the standard matrix for the linear transformation T: that first reflects points through the horizontal L-axis and then reflects points - through the vertical y-axis. 2. Show that the linear transformation T: R - R whose standard [ 2011 matrix is A= is onto but not one-to-one. - R$ whose standard 3. Show that the linear transformation T: R 0 1 matrix is A = 1 1 lov Lool is one-to-one...
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
Find the matrix [T], p of the linear transformation T: V - W with respect to the bases B and C of V and W, respectively. T:P, → P, defined by T(a + bx) = b - ax, B = {1 + x, 1 – x}, C = {1, x}, v = p(x) = 4 + 2x [T] C+B = Verify the theorem below for the vector v by computing T(v) directly and using the theorem. Let V and W...
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the result by 1 in the x-direction, -1 in the y-direction, and -1 in the z-direction, and then expands the y coordinate by a factor of 5 is given by A= The image of (5,-4,-1) under this transformation is , and Z = (1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the...