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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.6: Introduction To Linear Transformations

Problem 10EQ: In Exercises 7-10, give a counterexample to show that the given transformation is not a linear...

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Question

Suppose that T is a linear transformation such that
25
¹([2])-[2] ¹()-[13]
T
Write T as a matrix transformation.
T
For any ER2, the linear transformation T is given by I() =
U.

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Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
For the linear transformation from Exercise 33, find a T(1,1), b the preimage of (1,1), and c the preimage of (0,0). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRm by T(v)=Av. Find the dimensions of Rn andRm. A=[0110]
In Exercises 1-12, determine whether T is a linear transformation. 5. T:Mnn→ ℝ defined by T(A)=trt(A)
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 10.
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 7.
For the linear transformation from Exercise 37, find a T(1,0,2,3), and b the preimage of (0,0,0). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRm by T(v)=Av. Find the dimensions of Rn and Rm. A=[012114500131]
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 9.

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