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Exercise 5.6.9 Suppose T: R" → Rm is a linear transformation given by Tx=Ax where A is an m×n matrix. Show that T is never an isomorphism if m‡n. In particular, show that if m>n, T cannot be onto and if m

Exercise 5.6.9 Suppose T: R" → Rm is a linear transformation given by Tx=Ax where A is an m×n matrix. Show that T is never an isomorphism if m‡n. In particular, show that if m>n, T cannot be onto and if m

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
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**Exercise 5.6.9** Suppose \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a linear transformation given by \[ T\vec{x} = A\vec{x} \] where \( A \) is an \( m \times n \) matrix. Show that \( T \) is never an isomorphism if \( m \neq n \). In particular, show that if \( m > n \), \( T \) cannot be onto and if \( m < n \), then \( T \) cannot be one to one.
Transcribed Image Text:**Exercise 5.6.9** Suppose \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a linear transformation given by \[ T\vec{x} = A\vec{x} \] where \( A \) is an \( m \times n \) matrix. Show that \( T \) is never an isomorphism if \( m \neq n \). In particular, show that if \( m > n \), \( T \) cannot be onto and if \( m < n \), then \( T \) cannot be one to one.
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