3) Let T: R' R' be a linear transformation and X x2 T(x) is defined as X3 follows: T(x1,x2,x3) = (4.x1 – x2 +2x3, 3x1 +2x2 -5x3, 2x +x2 - X3) a) Find the standard matrix of T. b) Show that T is invertible by row reducing the appropriate matrix, and find the inverse of the standard matrix. c) Find a formula for T. T-v)= b. Give the equivalent

3) Let T: R' R' be a linear transformation and X x2 T(x) is defined as X3 follows: T(x1,x2,x3) = (4.x1 – x2 +2x3, 3x1 +2x2 -5x3, 2x +x2 - X3) a) Find the standard matrix of T. b) Show that T is invertible by row reducing the appropriate matrix, and find the inverse of the standard matrix. c) Find a formula for T. T-v)= b. Give the equivalent

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let T and T2 two linear transformations from R? to R², defined by 7(;) - ) ()-() 3x - 2y (;) (} x+ y T1 T2 y - 3x for all (x, y)" e R?. Find the standard matrix representations for each of the following composite functions, and their inverse transformation, if they exist, by using the algorithm for matrix inversion in Lemma 3, Topic 14. )T followed by T2. (i) T2 followed by an anticlockwise rotation by angle 5 in R?. (ii) A reflection in the line y = 2x followed by a projection onto the line y = -3x in R2.
Let T be a linear operator defined on P3 by T(a0+a1x+a2x^2)=a0+a1(x-1)+a2(x-1)^2. The third row of the matrix associated with T with respect to base 1, x, x^2 is: a) (0,0,1) b) other response c) (1,-2,1) d) (1,0,0) e) (-1,1,0)
5. Find the orthogonal projection of b = (1,-1,2) onto the left null space of the matrix A = -6
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8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,... are not vectors but are entries in vectors a) T(x1, x2) = (2x2-3x1, x1-4x2, 0, x2) b) T(x1, x2, x3, x4) = 2x1 + 3x - 4x4 (T:R* → R)
C: Let T be the linear operator associated in the standard basis B of F2 with the matrix 2 - [44] - 1 Let B' be the basis ((2, -1), (2, 1)) M = b) Using eachof these three matrices find the dimension of the range of T (you must check that the dimension is the same with all the matrices)
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Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
Could you explain how to show this in detail?
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