Question 2: Let V be a finite dimensional vector space over R, and let T: V → V be a linear map. Prove that if T is injective then T is surjective. a) HINT: Use the Rank-Nullity Theorem.
Question 2: Let V be a finite dimensional vector space over R, and let T: V → V be a linear map. Prove that if T is injective then T is surjective. a) HINT: Use the Rank-Nullity Theorem.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve problem 2
Transcribed Image Text:**Question 2:**
Let \( V \) be a finite dimensional vector space over \( \mathbb{R} \), and let \( T: V \rightarrow V \) be a linear map.
**a)** Prove that if \( T \) is injective then \( T \) is surjective.
**HINT:** Use the Rank-Nullity Theorem.
---
**b)** Prove that if \( T \) is surjective then \( T \) is injective.
![### Linear Algebra Problem
**Problem Statement:**
Let \( V \) be a vector space over \(\mathbb{R}\), and let \( T, S : V \to V \) be linear maps from \( V \) into \( V \). Suppose \( T \circ S \) is invertible. Prove that \( S \) is injective.
**Explanation:**
In the provided problem, \( V \) denotes a vector space over the real numbers \(\mathbb{R}\). The symbols \( T \) and \( S \) represent linear maps that take elements from the vector space \( V \) and map them back to \( V \). The objective is to prove that if the composition \( T \circ S \) (i.e., performing \( S \) first and then \( T \)) is an invertible linear transformation, then \( S \) is an injective (one-to-one) linear map.
**Solution Plan:**
To prove that \( S \) is injective, we need to show that if \( S(v) = S(w) \) for some vectors \( v, w \in V \), then \( v = w \).
1. **Assumption:**
\( T \circ S \) is invertible which implies there exists a linear map \( (T \circ S)^{-1} \) such that
\[
(T \circ S) \circ (T \circ S)^{-1} = \text{Id}_V
\]
where \(\text{Id}_V\) is the identity map on \( V \).
2. **Injectivity of \( S \):**
Suppose \( S(v) = S(w) \). We need to show \( v = w \).
3. **Action of \( T \) on \( S(v) \) and \( S(w) \):**
Given \( S(v) = S(w) \), applying \( T \) on both sides gives:
\[
T(S(v)) = T(S(w))
\]
4. **Using the Invertibility:**
Since \( T \circ S \) is invertible, we have:
\[
(T \circ S) \circ (T \circ S)^{-1} = \text{Id}_V
\]
Therefore,
\[
(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cf4b148-a9ff-447b-bf88-730c6d11afaf%2F73da649e-8965-4a5e-a0c9-eb94cfbc8a07%2Fogmtgkx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem
**Problem Statement:**
Let \( V \) be a vector space over \(\mathbb{R}\), and let \( T, S : V \to V \) be linear maps from \( V \) into \( V \). Suppose \( T \circ S \) is invertible. Prove that \( S \) is injective.
**Explanation:**
In the provided problem, \( V \) denotes a vector space over the real numbers \(\mathbb{R}\). The symbols \( T \) and \( S \) represent linear maps that take elements from the vector space \( V \) and map them back to \( V \). The objective is to prove that if the composition \( T \circ S \) (i.e., performing \( S \) first and then \( T \)) is an invertible linear transformation, then \( S \) is an injective (one-to-one) linear map.
**Solution Plan:**
To prove that \( S \) is injective, we need to show that if \( S(v) = S(w) \) for some vectors \( v, w \in V \), then \( v = w \).
1. **Assumption:**
\( T \circ S \) is invertible which implies there exists a linear map \( (T \circ S)^{-1} \) such that
\[
(T \circ S) \circ (T \circ S)^{-1} = \text{Id}_V
\]
where \(\text{Id}_V\) is the identity map on \( V \).
2. **Injectivity of \( S \):**
Suppose \( S(v) = S(w) \). We need to show \( v = w \).
3. **Action of \( T \) on \( S(v) \) and \( S(w) \):**
Given \( S(v) = S(w) \), applying \( T \) on both sides gives:
\[
T(S(v)) = T(S(w))
\]
4. **Using the Invertibility:**
Since \( T \circ S \) is invertible, we have:
\[
(T \circ S) \circ (T \circ S)^{-1} = \text{Id}_V
\]
Therefore,
\[
(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
