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.
The matrix of a linear transformation
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Differential Equations and Linear Algebra (4th Edition)
- 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_forwardThe figure shows three outline versions of the letter F. The second one is obtained from the first by shrinking horizontally by a factor of 0.75, and the third is obtained from the first by shearing horizontally by a factor of 0.25. a Find a data matrix D for the first letter F. b Find a transformation matrix T that transforms the first F into the second. Calculate TD, and verify that this is a data matrix for the second F. c Find the transformation matrix S that transforms the first F into the third. Calculate SD, and verify that this is a data matrix for the third F.arrow_forwardTrue or False? In Exercises 99-102, 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 Reflection that map a point in the xy-plane to its mirror image across the line y=x are linear transformations that are defined by the matrix [1001]. b Horizontal expansions or contractions are linear transformations that are defined by the matrix [k001].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_forwardTrue or False? In Exercises 99-102, 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 Reflection that map a point in the xy-plane to its mirror image across the x-axis are linear transformations that are defined by the matrix [1001]. b Vertical expansions or contractions are linear transformations that are defined by the matrix [100k].arrow_forwardCalculus Let B={1,x,x2,x3} be a basis for P3, and T:P3P4 be the linear transformation represented by T(xk)=0xtkdt. (a) Find the matrix A for T with respect to B and the standard basis for P4. (b) Use A to integrate p(x)=84x+3x3.arrow_forward
- 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.arrow_forwardThe Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y)=(5x+y,0,4x5y)arrow_forwardCalculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forward
- Let T:P2P3 be the linear transformation T(p)=xp. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3}.arrow_forwardTrue or false? det(A) is defined only for a square matrix A.arrow_forwardFinding the Standard Matrix and the Image In Exercise 11-22, a find the standard matrix A for the linear transformations T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the vector w=(3,1) in R2:T(v)=2projwvv, v=(1,4).arrow_forward
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