7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an ordered basis for V. Further, let TV → V be the linear operator defined as T(vn) := Oy, T(vi) := Vi+1, i=1,2,..., n-1. (a) What is [T]B.B? (b) Prove that T" = Ofun, but Tn-1 0fun. Remark on Notation: Note that T is a linear transformation, that is, a function. Here the notation T = ToToT, that is, T is a linear operator obtained by composing T with itself n times. Finally, n times Ofun VV is the zero linear operator, that is, it is a function that maps all elements of the domain into the zero vector in V. (c) Let S V→ V be a linear operator such that Sn = Ofun and Sn-10fun. Prove that there is an ordered basis for & for V such that the matrix representation [S], is equal to [T] B,B from part (a).
7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an ordered basis for V. Further, let TV → V be the linear operator defined as T(vn) := Oy, T(vi) := Vi+1, i=1,2,..., n-1. (a) What is [T]B.B? (b) Prove that T" = Ofun, but Tn-1 0fun. Remark on Notation: Note that T is a linear transformation, that is, a function. Here the notation T = ToToT, that is, T is a linear operator obtained by composing T with itself n times. Finally, n times Ofun VV is the zero linear operator, that is, it is a function that maps all elements of the domain into the zero vector in V. (c) Let S V→ V be a linear operator such that Sn = Ofun and Sn-10fun. Prove that there is an ordered basis for & for V such that the matrix representation [S], is equal to [T] B,B from part (a).
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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![7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an ordered basis for V.
Further, let T : V → V be the linear operator defined as
T(vn) := Oy,
T(vi) := Vi+1,
i=1,2,..., n - 1.
(a) What is [T]B,B ?
(b) Prove that In Ofun, but Tn-1 ‡ 0fun.
Remark on Notation: Note that T is a linear transformation, that is, a function. Here the notation
In :=
= ToToT, that is, T" is a linear operator obtained by composing T with itself n times. Finally,
n times
Ofun VV is the zero linear operator, that is, it is a function that maps all elements of the domain into the
zero vector in V.
(c) Let S : V → V be a linear operator such that Sn = 0fun and Sn-1 ‡0fun. Prove that there is an ordered
basis for & for V such that the matrix representation [S], is equal to [T] BB from part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2F9a92a733-5b1a-46c6-b472-05300cbb0f27%2F74cek7_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an ordered basis for V.
Further, let T : V → V be the linear operator defined as
T(vn) := Oy,
T(vi) := Vi+1,
i=1,2,..., n - 1.
(a) What is [T]B,B ?
(b) Prove that In Ofun, but Tn-1 ‡ 0fun.
Remark on Notation: Note that T is a linear transformation, that is, a function. Here the notation
In :=
= ToToT, that is, T" is a linear operator obtained by composing T with itself n times. Finally,
n times
Ofun VV is the zero linear operator, that is, it is a function that maps all elements of the domain into the
zero vector in V.
(c) Let S : V → V be a linear operator such that Sn = 0fun and Sn-1 ‡0fun. Prove that there is an ordered
basis for & for V such that the matrix representation [S], is equal to [T] BB from part (a).
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