1. Let L: R3 → M(2, 2) be a linear transformation such that (0-6). (f) -6...
Let L : R2 → R3 be a linear transformation such that L 1 1 = 1 2 3 and L 1 2 = 2 1 3 . Find L 2 1 Find the standard matrix representing L. Find the dimensions of the kernel and the range of L and their bases. 12. Let L : R² + RP be a linear transformation such that L | (3) - -(5)-(1) Find I (*) Find the standard matrix representing L. Find...
(1 point) Let f: R3 R3 be the linear transformation defined by f(3) = [ 2 1 1-4 -2 -57 -5 -4 7. 0 -2 Let B C = = {(2,1, -1),(-2,-2,1),(-1, -2, 1)}, {(-1,1,1),(1, -2, -1),(-1,3, 2)}, be two different bases for R. Find the matrix (fls for f relative to the basis B in the domain and C in the codomain. [] =
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x, y, z) = (–2x + 2y +z, -x +2y). Find the matrix that corresponds to f with respect to the canonical bases of R3 and R2.
-00)0) 2 (AB 22) Let L : R, R2 be a linear transformation. You are given that L 2- 3 (a) Find the matrix A that represents L with respect to the basisu-| | 2-1 1-1 4 1 and the 6 standard basis F1 (b) Find the matrix B that represents IL with respect to the standard basis in both R3 and R2
1. Let F: R4-R3 be a linear transformation satisfying F(1,1,1,1) (0, 1,2), F(1,1,0, 1)(0, 0,2) F(0,1,0, 0) 1,0,0) F(1,1,0,0) (0,0,0), (a) Calculate F(x, y, z, w) (b) Calculate ker(F) and R(F)
Let L:R3→R3 be a linear transformation. If L is the rescaling transformation by the factor 5, then L(15,−16,−24) = ( , , ) If L projects R3 orthogonally onto the xy-plane, then L(15,−16,−24) = ( , , ) If L projects R3 orthogonally onto the xz-plane, then L(15,−16,−24) = ( , , ) If L RHR-rotates R3 about the positively oriented y-axis through π / 23 , then ‖L(15,−16,−24)‖^2 = _____ If L is an isometry, then ‖L(15,−16,−24)‖^2 = _____
Let ?: ?2(R) ⟶ R3 be a linear transformation such that ?(1) = (−1, 2, −3), ?(1 + 3?) = (4, −5, 6), and ?(1 + ?2) = (−7, 8, −9). a. Show that {1,1 + 3? ,1 + ?2} is a basis for ??2(R) (7pts) b. Compute ?(−1 + 4? + 2?2). (3pts)
2. (5 points) Let T: R2 + R3 be a linear transformation with 2x1 - x2] 1-3x1 + x2 | 2x1 – 3x2 Find x = (x) <R? such that [0] -1 T(x) = (-4)
(2 points) Let -1 7 A = -9 5 -8 -6 a R3 by T(x Aï. Find the images of u Define the linear transformation T : R- and y 4 under = - T. T(M TM = (2 points) Let -1 7 A = -9 5 -8 -6 a R3 by T(x Aï. Find the images of u Define the linear transformation T : R- and y 4 under = - T. T(M TM =
Let T: R3 → R3 be the linear transformation that projects u onto v = (9, -1, 1). (a) Find the rank and nullity of T. rank nullity (b) Find a basis for the kernel of T.