Let T₁: V₁ Consider the following questions. A. Prove that if T₁ and T₂ are both one-to-one, then so is T₂T₁: V₁ → V3 B. Prove that if T₁ and T₂ are both onto, then so is T₂T₁1: V₁ → V3 C. Prove that if T₁ and T₂ are both isomorphisms, then so is T₂T₁: V₁ →→ V3 V₂ and T2: V₂ V3 be linear transformations. The discussion is to determine the answer to the following. If you can do a proof for them, then use proof.
Let T₁: V₁ Consider the following questions. A. Prove that if T₁ and T₂ are both one-to-one, then so is T₂T₁: V₁ → V3 B. Prove that if T₁ and T₂ are both onto, then so is T₂T₁1: V₁ → V3 C. Prove that if T₁ and T₂ are both isomorphisms, then so is T₂T₁: V₁ →→ V3 V₂ and T2: V₂ V3 be linear transformations. The discussion is to determine the answer to the following. If you can do a proof for them, then use proof.
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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READ THE FOLLOWING INSTRUCTIONS FIRST:
Can you please solve the problem on the picture, Show all of your work and explain each step. PLEASE POST PICTURES OF YOUR WORK AND DO NOT TYPE IT, IT IS HARDER TO UNDERSTAND WHEN YOU TYPE IT! THANK YOU.
Transcribed Image Text:**Initial Post Prompt**
Let
\( T_1 : V_1 \rightarrow V_2 \) and \( T_2 : V_2 \rightarrow V_3 \) be linear transformations.
Consider the following questions. The discussion is to determine the answer to the following. If you can do a proof for them, then use proof.
A. Prove that if \( T_1 \) and \( T_2 \) are both one-to-one, then so is \( T_2 T_1 : V_1 \rightarrow V_3 \).
B. Prove that if \( T_1 \) and \( T_2 \) are both onto, then so is \( T_2 T_1 : V_1 \rightarrow V_3 \).
C. Prove that if \( T_1 \) and \( T_2 \) are both isomorphisms, then so is \( T_2 T_1 : V_1 \rightarrow V_3 \).
You should use the definitions of one-to-one and onto, isomorphism, Kernel, Range. You will also want to consider the co-domains and domains for the transformations. Think about transformations and their impacts.
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this is a problem from another classmate, Can you please look at his work and identify there is any mistakes. write three big comments on how he should solve the problem and what would be easier, include theromes as well. THank you
POST PICTURES OF YOUR WORK PLEASE, DO NOT TYPE IT!
![**Linear Transformations and Isomorphisms**
Let \(T_1: V_1 \rightarrow V_2\) and \(T_2: V_2 \rightarrow V_3\) be linear transformations.
Consider the following questions. The discussion aims to determine the answers to the following. If you can provide a proof for them, then use proof.
A. Prove that if \(T_1\) and \(T_2\) are both one-to-one, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
B. Prove that if \(T_1\) and \(T_2\) are both onto, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
C. Prove that if \(T_1\) and \(T_2\) are both isomorphisms, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
### Proofs:
**a) One-To-One:**
Let \(v_1, v_2 \in V_1\), \(v_1 \neq v_2\).
If \(T_1\) is one-to-one,
\[ T_1(v_1) \neq T_1(v_2) \]
If \(T_1(v_1) \neq T_1(v_2) \in V_2 \), and \(T_2\) is one-to-one:
\[ T_2(T_1(v_1)) \neq T_2(T_1(v_2)) \]
Therefore, \(T_2 \circ T_1(v_1) \neq T_2 \circ T_1(v_2)\).
Hence, \(T_1\) and \(T_2\) are both one-to-one.
**b) Onto:**
For every \(v_3 \in V_3\),
\(T_2\) is onto implies there exists \(v_2 \in V_2\) such that \(T_2(v_2) = v_3\).
\(T_1\) is onto implies for every \(v_2 \in V_2\), there exists \(v_1 \in V_1\) such that \(T](https://content.bartleby.com/qna-images/question/0655393b-8df3-4633-b13c-e0d6983d2306/6153f7b5-254e-4f46-a87b-8df42841f90e/7c5jbi_processed.png)
Transcribed Image Text:**Linear Transformations and Isomorphisms**
Let \(T_1: V_1 \rightarrow V_2\) and \(T_2: V_2 \rightarrow V_3\) be linear transformations.
Consider the following questions. The discussion aims to determine the answers to the following. If you can provide a proof for them, then use proof.
A. Prove that if \(T_1\) and \(T_2\) are both one-to-one, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
B. Prove that if \(T_1\) and \(T_2\) are both onto, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
C. Prove that if \(T_1\) and \(T_2\) are both isomorphisms, then so is \(T_2 \circ T_1: V_1 \rightarrow V_3\).
### Proofs:
**a) One-To-One:**
Let \(v_1, v_2 \in V_1\), \(v_1 \neq v_2\).
If \(T_1\) is one-to-one,
\[ T_1(v_1) \neq T_1(v_2) \]
If \(T_1(v_1) \neq T_1(v_2) \in V_2 \), and \(T_2\) is one-to-one:
\[ T_2(T_1(v_1)) \neq T_2(T_1(v_2)) \]
Therefore, \(T_2 \circ T_1(v_1) \neq T_2 \circ T_1(v_2)\).
Hence, \(T_1\) and \(T_2\) are both one-to-one.
**b) Onto:**
For every \(v_3 \in V_3\),
\(T_2\) is onto implies there exists \(v_2 \in V_2\) such that \(T_2(v_2) = v_3\).
\(T_1\) is onto implies for every \(v_2 \in V_2\), there exists \(v_1 \in V_1\) such that \(T
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