7.8.Show that T(x1, x2, x3) = (3x₁ - 6x3, x2 + x3, X1₁ - X2 - 2x3) is a linear trans-formation by finding a matrix that implements the mapping. Note that x₁, x2, and x3are not vectors but are entries in vectors.Determine if the linear transformation from the previous problem is(a) onto(b) one-to-one

We use definition of linear transformation and properties of linear transformation.

Answered: 7. 8. Show that T(x1, x2, x3) = (3x₁ −…

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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2. The Stretching Problem Imagine a linear transformation T: R² →R² that has the following properties: In the direction along the line y = -3x, the transformation stretches all points by a factor of two. In the direction along the line y = x, the transformation keeps all points fixed. 2a).Use the space on the right to sketch what should happen to the image shown on the left when it is stretched according to the transformation described above. You may use a combination of intuition or calculations, as well as any additional sketches below or on your group's whiteboard. IN
4a). The transformation defined by the matrix A = [14 5 stretches images in R² in the directions y = x and y = -x. Figure out the factor by which anything in the y = x direction is stretched and the factor by which anything in the y = -x direction is stretched. 4b). The transformation defined by the matrix B = -82 -55 13 3] stretches images in R2 in one direction by a factor of 3 and some other direction by a factor of 2. Figure out what direction gets stretched by a factor of 3 and what direction gets stretched by a factor of 2. 4c). The transformation defined by the matrix C = stretches images in R2 in two directions. Find the directions and the factors by which it stretches in those directions.
5. Determine the matrix of the transformation that first reflects points through the horizontal x-axis and then reflects points through the line y = x.
The matrices A₁ = A3 [] = ^--^- A₂ = [] =[] A4 form a basis for the linear space V = R 2x2. Write the matrix of the linear transformation T: R 2x2R 2x2 such that T(A) = 3A + 10AT relative to this basis.
5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.
Suppose M is a linear transformation taking vectors in R² to vectors in R2, where M(1,0)=(2,-4) M(0, 1) = (-6,6) M(2,9) = (
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)
5. 3 (a) Let A To which point does A map the point (2, –5)? To which point does A map the point (1,0)? 1 (b) Let B = To which point does B map the point (11, 8)? To 3 which point does B map the point (3, 4)? (c) Find the matrix which maps the point (2, –5) to the point (19, 13) and the point (1,0) to (7,9).
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Show that T is a linear transformation by finding a matrix that implements the mapping. Note that X₁, X2.. vectors but are entries in vectors. T(X1 X2 X3) = (x1 - 3x2 + 2x3, x2 - 9x3) A= (Type an integer or decimal for each matrix element.) are not
Suppose T: M2,2→→P2 is a linear transformation whose action on a basis for M2,2 is as follows: [18]-- Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the character to indicate an exponent, e.g. ax^2-bx+c. = -2x+5 T = 0 1 1 -2 -2 = -x+4 T -4 -2 = 2x² +4x-26 T 00 = -2x²+2x+10

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