Definition 1 Let T R →→ Rm be a linear transformation. • We say T is exact if { 7 € R¹ : T(T) = õ} = {0}. • We say T is comprehensive if for every y € Rm there is at least one ☞ € R" such that T(x) = ỹ. (a) Give an example of an exact linear transformation T: R² → R² such that At has no eigenvectors. (b) Let T: R → R" be a linear transformation. Show that T is exact if and only if AȚ is invertible. Note. To show that an "if and only if" statement is true, you have to show that each statement implies the other. (c) Show that T is comprehensive if and only if span {T(ē₁), T(2),..., T(en)} = Rm. Hint. The result you proved on Written Assignment 1 might be helpful again here! (d) True or false: If T: R" → Rm is comprehensive, then it is exact. (e) True or false: If n > m, then any linear transformation T: Rn → Rm is not exact.
Definition 1 Let T R →→ Rm be a linear transformation. • We say T is exact if { 7 € R¹ : T(T) = õ} = {0}. • We say T is comprehensive if for every y € Rm there is at least one ☞ € R" such that T(x) = ỹ. (a) Give an example of an exact linear transformation T: R² → R² such that At has no eigenvectors. (b) Let T: R → R" be a linear transformation. Show that T is exact if and only if AȚ is invertible. Note. To show that an "if and only if" statement is true, you have to show that each statement implies the other. (c) Show that T is comprehensive if and only if span {T(ē₁), T(2),..., T(en)} = Rm. Hint. The result you proved on Written Assignment 1 might be helpful again here! (d) True or false: If T: R" → Rm is comprehensive, then it is exact. (e) True or false: If n > m, then any linear transformation T: Rn → Rm is not exact.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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