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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 15CM

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Determine whether the linear transformation is invertible. If it is, finds its inverse.

T(x,y,z,w) = (x-3y, y, z+w, z)

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Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find T(7,2).
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A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x,y)=(x2,y+1), determine the images of (0,0,),(2,1), and (5,4). (c) Show that a translation in R2 has no fixed points.
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For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.
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A linear transformation T : R2 → R2first reflects points through the x1-axis andthen reflects points through the line x2 = −x1. Show that T can also be describedas a linear transformation that rotates points about the origin. What is the angleof that rotation?
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Find the Jacobian of the transformation x = 5u – 5v, y = u² − 2v.

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