assume that T is a lineartransformation. Find aStandard matrix of T.T: IR² →IR"8)whereAT(e₁) = (5,1,5,1)T(e₂) = (-3₁7,0,0)?e₁ = (1,0)€ 2 = (0₁1)(integer or decimal)

Answered: Image /qna-images/answer/feb09744-e96d-41fb-b824-81f9ca0b76d2.jpg

Answered: 8) assume that I is a linear…

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 v be a vector in R" and define a function T, : R" → R by T,(x) = x · v. (a) Show that T, is a linear transformation. (b) Find the standard matrix for Ty, where v = = (1, 1, 1).
Let B = {-3,2), (4, -2)} and B' = {(-1,2), (2,-2)} a. Find the transition matrix from B to B'. b. Find the coordinate matrix [x]B given that [x]B -G]-
10. Let T be the linear transformation from R3 to R$ that rotates every vector around the y-axis by radians, counterclockwise as viewed from the point (0,5, 0), and afterwards multiplies its length by 4. (a) Find the matrix of transformation T. (b) Calculate T((1, 2, 3)).
The cross product of two vectors in R' is defined by 日-屏 [b1 azb3 – azb2 ai a2 b2 azbı – aıb3 b3 aıb2 – azbı аз Let i . Find the matrix A of the linear transformation from R to R° given by T(7) = j x a. A = || ||
Consider a linear transformation T from R³ to R2 for which 0 3 6 T ([]) - · ~ ([1]) - · · (8) - C] = [{}], T = 8 2 T 0 = 5 Find the matrix A of T. A = =
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁0 T₂. T₁: R² R², T₁(x, y) = (x - 3y, 3x + 3y) T₂: R² → R², T₂(x, y) = (y, 0) A = A' = ↓ ↑ 00
Let S be a linear transformation from R2 to R2 with associated matrix 11. Let T be a linear transformation from R2 to R2 with associated matrix -F A = C= B = 2 Determine the matrix C of the composition To S. -3 2] 0 2
Find the matrix of the linear transformation T from R2 - R² where T([1, 0]) = [1, –2] and T([2, 1]) = [2, 3] %3D (Note: matrix A = [T([1, 0]), T([0, 1])] 1 -2 (a) 1 (b) 1 (c) -2 (d) -2 3 7 3 Ob a C
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 T is a linear transformation, where (1, 0, 0) T(u) T(7) (0, 1,0) W (0,0,1) T(w) Then I can be represented by the matrix ū V = = = T= = = = (5, -3, -2) (-2,-1,0) (0,5, 4)
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)
5) PLEASE ANSWER EACH QUESTION, THANKS.

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8) assume that I is a linear transformation. Find a Standard matrix of T. T: IR ² IR4 where T(e) = (5,1,5,1) T(e₂) = (-3₁7,0,0) e₁ = (1,0)