Let FR" → R" and be a linear transformation. Suppose the matrix representation of F with respect to the standard basis B := {₁,..., en} CR" is an n x n matrix A. Show that the following linear transformation F(³): R¹ →R", F3³)(v):= F(F(F(v))) has matrix representation with respect to the basis B given by the matrix A³. Justify your answer.
Let FR" → R" and be a linear transformation. Suppose the matrix representation of F with respect to the standard basis B := {₁,..., en} CR" is an n x n matrix A. Show that the following linear transformation F(³): R¹ →R", F3³)(v):= F(F(F(v))) has matrix representation with respect to the basis B given by the matrix A³. Justify your answer.
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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![**Linear Transformation and Matrix Representation**
Let \( F : \mathbb{R}^n \to \mathbb{R}^n \) be a linear transformation. Suppose the matrix representation of \( F \) with respect to the standard basis \( B := \{ e_1, \ldots, e_n \} \subset \mathbb{R}^n \) is an \( n \times n \) matrix \( A \). Show that the following linear transformation
\[
F^{(3)} : \mathbb{R}^n \to \mathbb{R}^n, \quad F^{(3)}(v) := F(F(F(v)))
\]
has a matrix representation with respect to the basis \( B \) given by the matrix \( A^3 \). Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fb02365-efda-4f71-9185-5332218a7cc4%2F49a7796f-47e3-4941-b117-2b763a0f0245%2Fhxr0zgl_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Transformation and Matrix Representation**
Let \( F : \mathbb{R}^n \to \mathbb{R}^n \) be a linear transformation. Suppose the matrix representation of \( F \) with respect to the standard basis \( B := \{ e_1, \ldots, e_n \} \subset \mathbb{R}^n \) is an \( n \times n \) matrix \( A \). Show that the following linear transformation
\[
F^{(3)} : \mathbb{R}^n \to \mathbb{R}^n, \quad F^{(3)}(v) := F(F(F(v)))
\]
has a matrix representation with respect to the basis \( B \) given by the matrix \( A^3 \). Justify your answer.
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