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Chapter 1, Problem 18E(choose chapter or problem)
PROBLEM 18EIn Exercises, 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.T(x1, x2) = (2x2? 3x1, x1 ? 4x2,0, x2)
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QUESTION:
PROBLEM 18EIn Exercises, 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.T(x1, x2) = (2x2? 3x1, x1 ? 4x2,0, x2)
ANSWER:PROBLEM 18EIn Exercises, 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.T(x1, x2) = (2x2 3x1, x1 4x2,0, x2)Solution :Step 1: We have to prove that T is linear transformation and we have to find a matrix that implements the mapping.Linear Transformation : The function T : Rn Rn is called the linear transformation by the formula T(x) = Ax, where A is an n × n matrix.1. Suppose u and v are vectors in Rn . Then T(u + v) = A(u + v) = Au + Av = T(u) + T(v). 2. Suppose u is a vector in Rn and c is a scalar. Then T(cu) = A(cu) = cAu = cT(u).