Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 4EQ
See similar textbooks
icon
Related questions
Question

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.

Initial Post Prompt Let T₁: V₁ → V₂ and T2: V₂ → V3 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 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 T2T₁: V₁ → V3 C. Prove that if T₁ and T₂ are both isomorphisms, then so is T₂T₁: V₁ → V3 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
Transcribed Image Text:Initial Post Prompt Let T₁: V₁ → V₂ and T2: V₂ → V3 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 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 T2T₁: V₁ → V3 C. Prove that if T₁ and T₂ are both isomorphisms, then so is T₂T₁: V₁ → V3 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
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

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!

Let T₁: V₁ V₂ and T₂: V₂ V3 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 T₁ and T2 are both one-to-one, then so is T2T1₁: V₁ → V3 B. Prove that if T₁ and T2 are both onto, then so is T2T1: V₁ → V3 C. Prove that if T₁ and T2 are both isomorphisms, then so is T2T1: V1 V3 T₁: V₁ V₂ T₂: V₂ V3 a) Let V₁, V₂ EV, V, ± √₂ If T, is one to one → T(V₁) T₁ (1₂) T(M), T₁ (V₂) € V 1/₂ is one to one → ) - 1₁ & 1₂ are both one to one b) √₂ € V₂. 1₂ onto =) V₂ € V₂ Ti onto, Vz EV ₂ =) T₂ T₁ (y) = T₂T₁ (v₂) T₂ (V₂) = 13. VIEV T₂ ( Tj (v₁) # T₂ (T₁(V₂)) -) T₁ (V₁) = √₂ → Rng [ 1₂T₁ (V)] = Rng [T₂ (T₁ (U))] = Rng T₂ (R ng TI (V)] = Rng [T2₂ (1₂1] = √3 d c) T₁ & 1₂ are Isomorphisms when TI & T2 are both one to one, and onto ( proof on part a&b) T₁, T₂ is a linear transformation Let is a scalar V₁, V₂ EV, T₂T₁ (V₁ + V₂) = T₂ (T₁ (v₁ +V₂)) = T₂ (T₁ (v₁) + T₁ (v₂) = T₂Tilvi)+ Tz Tilv₂) (T₂T₁v₁ + (T₂T₁) V₂ T₂ T1 (KV) T₂ T₁ (α v) = T₂ (αT₁ (v)) = α T₂ T₁ (v) = α (T₂T₁)(v) T₂, T₁ is a linear transformation -) T₂, T₁ is an isomorphism.
Transcribed Image Text:Let T₁: V₁ V₂ and T₂: V₂ V3 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 T₁ and T2 are both one-to-one, then so is T2T1₁: V₁ → V3 B. Prove that if T₁ and T2 are both onto, then so is T2T1: V₁ → V3 C. Prove that if T₁ and T2 are both isomorphisms, then so is T2T1: V1 V3 T₁: V₁ V₂ T₂: V₂ V3 a) Let V₁, V₂ EV, V, ± √₂ If T, is one to one → T(V₁) T₁ (1₂) T(M), T₁ (V₂) € V 1/₂ is one to one → ) - 1₁ & 1₂ are both one to one b) √₂ € V₂. 1₂ onto =) V₂ € V₂ Ti onto, Vz EV ₂ =) T₂ T₁ (y) = T₂T₁ (v₂) T₂ (V₂) = 13. VIEV T₂ ( Tj (v₁) # T₂ (T₁(V₂)) -) T₁ (V₁) = √₂ → Rng [ 1₂T₁ (V)] = Rng [T₂ (T₁ (U))] = Rng T₂ (R ng TI (V)] = Rng [T2₂ (1₂1] = √3 d c) T₁ & 1₂ are Isomorphisms when TI & T2 are both one to one, and onto ( proof on part a&b) T₁, T₂ is a linear transformation Let is a scalar V₁, V₂ EV, T₂T₁ (V₁ + V₂) = T₂ (T₁ (v₁ +V₂)) = T₂ (T₁ (v₁) + T₁ (v₂) = T₂Tilvi)+ Tz Tilv₂) (T₂T₁v₁ + (T₂T₁) V₂ T₂ T1 (KV) T₂ T₁ (α v) = T₂ (αT₁ (v)) = α T₂ T₁ (v) = α (T₂T₁)(v) T₂, T₁ is a linear transformation -) T₂, T₁ is an isomorphism.
Solution
Bartleby Expert
SEE SOLUTION
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS