True-False Review
For Questions (a)-(f), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
A linear transformation
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- True or False? In Exercises 53 and 54, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If T:RnRm is a linear transformation such that T(e1)=[a11,a21am1]TT(e2)=[a12,a22am2]TT(en)=[a1n,a2namn]T then the mn matrix A=[aij] whose columns corresponds to T(ei) is such that T(v)=Av for every v in Rn is called the standard matrix for T. b All linear transformations T have a unique inverse T1.arrow_forwardCAPSTONE Let T:R4R3 be the linear transformation represented by T(x)=Ax, where A=[121001230001]. (a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T is onto? Explain. (f) Is T an isomorphism? Explain.arrow_forwardLinear TransformationsIn Exercises 9-22, determine whether the function is a linear transformation. T:33, T(x,y,z)=(x+y,xy,z)arrow_forward
- Guided Proof Let B be an invertible nn matrix. Prove that the linear transformation T:Mn,nMn,n represented by T(A)=AB is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} Theorem 6.6. Use the fact that B is an invertible nn matrix and that T(A)=AB. (iii) Conclude that T is an isomorphism.arrow_forwardFinding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T)and d rank(T). A=[494929494929292919]arrow_forwardA 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.arrow_forward
- Linear TransformationsIn Exercises 9-22, determine whether the function is a linear transformation. T:33, T(x,y,z)=(x+1,y+1,z+1)arrow_forwardFinding the Inverse of a Linear Transformation In Exercise 49-52, determine whether the linear transformation is invertible. If it is, find its inverse. T:R2R2, T(x,y)=(x,y)arrow_forwardFinding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[111201]arrow_forward
- In Exercises 1-3, fill in the blanks. A nonrigid transformation of the graph of y=fx represented by gx=cfx is a when c1 and a when 0c1.arrow_forwardFinding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[910310310110]arrow_forwardCalculus In Exercises 61-64, for the linear transformation from Example 10, find the preimage of each function. Dx(f)=sinxarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning