Category Archives: linear algebra

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 →

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 →

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 →

Exercise 2.6.15. Suppose that is a linear transformation from to itself, or more generally from any vector space to itself. Show that is also a linear transformation. Answer: If is a linear transformation from some vector space to itself then … Continue reading →

Exercise 2.6.14. Suppose that , , and are linear transformations, with taking vectors from to , taking vectors from to , and  taking vectors from to . Consider the product of these transformations. It starts with a vector in and … Continue reading →

Exercise 2.6.13. Suppose that is a linear transformation from the – plane to itself. If a transformation exists such that show that is also linear. Also show that if is the matrix representing then the matrix representing must be . … Continue reading →

Exercise 2.6.12. If is the reflection matrix in the – plane, show that using the trigonometric identity ( for short). Answer: We have so that Since this can be simplified to NOTE: This continues a series of posts containing worked … Continue reading →

Exercise 2.6.11. Consider the vector space of functions in for which . From the previous exercise we can express any such function as where and are basis vectors for W. Suppose at we have and . Find and such that … Continue reading →

Exercise 2.6.10. Consider the vector space of functions in for which . Find two functions that can serve as basic vectors for the space. Answer: For a function to be in the space its second derivative with respect to must … Continue reading →

Exercise 2.6.9. Considering taking a polynomial from , the space of cubic polynomials in , and multiplying it by the polynomial to produce a polynomial in , the space of polynomials in of degree four. Describe a matrix representing this multiplication … Continue reading →