Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
one? 37. Suppose T and U are linear transformations from R" to R" such that T (UX) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not?

one? 37. Suppose T and U are linear transformations from R" to R" such that T (UX) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

37

Invertible Matrix Theorem, expl Ax = b must have a solution for each b in R". In Exercises 33 and 34, T is a linear transformation from R2 into R2. Show that T is invertible and find a formula for 7-¹. 33. T(x₁, x₂) = (-5x₁ + 9x2,4x1 - 7x2) 34. T(x₁, x₂) = (6x₁8x2, -5x1 + 7x₂) X2) 35. Let T: "R" be an invertible linear transformation. Ex- plain why T is both one-to-one and onto R". Use equations (1) and (2). Then give a second explanation using one or more theorems. lo enm 36. Let T be a linear transformation that maps R" onto R". Show that T exists and maps R" onto R". Is T-¹ also one-to- -1 -1 one? X 37. Suppose I and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(TX) = x for all x in R"? Why or why not? T 38. Suppose a linear transformation 7 : R" → R" has the prop- erty that T(u) = T(v) for some pair of distinct vectors u and v in R". Can T map R" onto R"? Why or why not? nottoto till som BR Pups parl n 39. Let T: R" → R" be an invertible linear transformation, and let S and U be functions from R" into R" such that S (T(x)) = x and U (T(x)) = x for all x in R". Show that U(v) = S(v) for all v in R". This will show that I has a unique inverse, as asserted in Theorem 9. [Hint: Given any v in R", we can write v = T(x) for some x. Why? Compute S(v) and U(v).] On toivre sur 40. Suppose T and S satisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that S is a linear transformation. [Hint: Given u, v in R", let x = S(u), y = S(v). Then T(x) = u, T(y) = v. Why? Apply S to both sides of the equation T(x) + T(y) = T(x + y). Also, consider T(cx) = cT(x).] c. Use your m ber of the c Exercises 42-44 sh trix A to estimate th If the entries of A and if the conditio positive integer), t usually be accurate 42. [M] Find the Construct ar Then use you of Ax= b. T the number c and report he used in place 43. [M] Repeat 44. [M] Solve a column of th A = 1 1/2 1/3 1/4 1/5 How many correct? Ex 56700,-88 45. [M] Some mand to cre use an inve order or la what you fi Master 10 11 SOLUTIONS TO PRACTICE PROBLEM 1. The columns of A are ohvioual SG
Transcribed Image Text:Invertible Matrix Theorem, expl Ax = b must have a solution for each b in R". In Exercises 33 and 34, T is a linear transformation from R2 into R2. Show that T is invertible and find a formula for 7-¹. 33. T(x₁, x₂) = (-5x₁ + 9x2,4x1 - 7x2) 34. T(x₁, x₂) = (6x₁8x2, -5x1 + 7x₂) X2) 35. Let T: "R" be an invertible linear transformation. Ex- plain why T is both one-to-one and onto R". Use equations (1) and (2). Then give a second explanation using one or more theorems. lo enm 36. Let T be a linear transformation that maps R" onto R". Show that T exists and maps R" onto R". Is T-¹ also one-to- -1 -1 one? X 37. Suppose I and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(TX) = x for all x in R"? Why or why not? T 38. Suppose a linear transformation 7 : R" → R" has the prop- erty that T(u) = T(v) for some pair of distinct vectors u and v in R". Can T map R" onto R"? Why or why not? nottoto till som BR Pups parl n 39. Let T: R" → R" be an invertible linear transformation, and let S and U be functions from R" into R" such that S (T(x)) = x and U (T(x)) = x for all x in R". Show that U(v) = S(v) for all v in R". This will show that I has a unique inverse, as asserted in Theorem 9. [Hint: Given any v in R", we can write v = T(x) for some x. Why? Compute S(v) and U(v).] On toivre sur 40. Suppose T and S satisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that S is a linear transformation. [Hint: Given u, v in R", let x = S(u), y = S(v). Then T(x) = u, T(y) = v. Why? Apply S to both sides of the equation T(x) + T(y) = T(x + y). Also, consider T(cx) = cT(x).] c. Use your m ber of the c Exercises 42-44 sh trix A to estimate th If the entries of A and if the conditio positive integer), t usually be accurate 42. [M] Find the Construct ar Then use you of Ax= b. T the number c and report he used in place 43. [M] Repeat 44. [M] Solve a column of th A = 1 1/2 1/3 1/4 1/5 How many correct? Ex 56700,-88 45. [M] Some mand to cre use an inve order or la what you fi Master 10 11 SOLUTIONS TO PRACTICE PROBLEM 1. The columns of A are ohvioual SG
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS