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Monthly Archives: September 2013
Linear Algebra and Its Applications, Exercise 2.6.21
Exercise 2.6.21. Consider the transformation from to that takes into . What is the axis of rotation for the transformation? What is the angle of rotation? Answer: We can approach this problem in at least two ways. The first way … Continue reading
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Linear Algebra and Its Applications, Exercise 2.6.20
Exercise 2.6.20. A nonlinear transformation from a vector space to a vector space is invertible a) if for any in there exists some in such that and b) if and are in then implies that . Describe which of the … Continue reading
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Linear Algebra and Its Applications, Exercise 2.6.19
Exercise 2.6.19. Let be the vector space consisting of all cubic polynomials of the form and let be the subset of consisting of only those cubic polynomials for which . Show that is a subspace of and find a set … Continue reading
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Linear Algebra and Its Applications, Exercise 2.6.18
Exercise 2.6.18. Given a vector in find a matrix that produces a corresponding vector in in which all entries are shifted right one place. Find a second matrix that takes a vector in and produces the vector in in which … Continue reading
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Linear Algebra and Its Applications, Exercise 2.6.17
Exercise 2.6.17. Find a matrix corresponding to the linear transformation of cyclically permuting vectors in such that applied to produces . Determine the effect of and and explain why . Answer: We can construct by considering its effect on the … Continue reading
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Linear Algebra and Its Applications, Exercise 2.6.16
Exercise 2.6.16. Consider the space of 2 by 2 matrices. Any such matrix can be represented as the linear combination of the matrices that serve as a basis for the space. Find a matrix corresponding to the linear transformation of … Continue reading
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The composition of linear transformations is a linear transformation
In doing the answers to exercise 2.6.14 in Gilbert Strang’s Linear Algebra and Its Applications, Third Edition I noticed one of the downsides of the book: While Strang’s focus on practical applications is usually welcome, sometimes in his desire to … Continue reading
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