6. Space V is the space of all polynomials of degree 3: V = {f(x) = ax³ + bx² + cx + d}. Space W is defined as all those polynomials ƒ from V such that f(1) = 0: W = {f € V : f(1) = 0} %3D Question a. Prove that space W is a linear subspace of V.
6. Space V is the space of all polynomials of degree 3: V = {f(x) = ax³ + bx² + cx + d}. Space W is defined as all those polynomials ƒ from V such that f(1) = 0: W = {f € V : f(1) = 0} %3D Question a. Prove that space W is a linear subspace of V.
Intermediate Algebra
10th Edition
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Functions
Section9.1: Relations And Functions
Problem 75PS
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Please solve only question a in complete English sentence.
![Below is the transcribed text for an educational website, with detailed explanations of any mathematical concepts, graphs, or diagrams.
---
### Linear Algebra Question on Polynomial Spaces and Linear Transformations
#### Problem Statement
We are given the following scenario:
1. **Space \( V \) Definition:**
Space \( V \) is the space of all polynomials of degree 3.
\[
V = \{ f(x) = ax^3 + bx^2 + cx + d \}
\]
2. **Space \( W \) Definition:**
Space \( W \) is defined as all those polynomials \( f \) from \( V \) such that \( f(1) = 0 \).
\[
W = \{ f \in V : f(1) = 0 \}
\]
3. **Question a:**
Prove that space \( W \) is a linear subspace of \( V \).
4. **Question b:**
Let \( D(f(x)) \) be a transformation of \( V \) defined as follows:
\[
D(f(x)) = f'(x)
\]
Prove that \( D \) is a linear transformation. Note that \( D \) is a derivative of \( f \).
5. **Question c:**
Find the matrix of \( D \) in the basis \( \vec{v}_1 = x^3 \), \( \vec{v}_2 = x^2 \), \( \vec{v}_3 = x \), \( \vec{v}_4 = 1 \). Then, find its kernel and range.
---
This problem requires students to demonstrate their understanding of polynomial spaces, subspaces, linear transformations, and matrix representation of these transformations.
**Explanation of Key Concepts:**
1. **Polynomial Spaces:**
A polynomial space of degree 3 consists of all polynomials where the highest degree term is \( x^3 \). For example, \( f(x) = ax^3 + bx^2 + cx + d \).
2. **Subspaces:**
A subspace is a subset of a vector space that itself forms a vector space under the same operations of addition and scalar multiplication.
3. **Linear Transformations:**
A transformation \( D \) is linear if it satisfies additivity and homogeneity:
- Additivity](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff003da25-8c7d-4491-8329-a317b002c309%2F68b22e8a-a22b-4a8d-96cc-bed3f994a426%2F87z4hav_processed.png&w=3840&q=75)
Transcribed Image Text:Below is the transcribed text for an educational website, with detailed explanations of any mathematical concepts, graphs, or diagrams.
---
### Linear Algebra Question on Polynomial Spaces and Linear Transformations
#### Problem Statement
We are given the following scenario:
1. **Space \( V \) Definition:**
Space \( V \) is the space of all polynomials of degree 3.
\[
V = \{ f(x) = ax^3 + bx^2 + cx + d \}
\]
2. **Space \( W \) Definition:**
Space \( W \) is defined as all those polynomials \( f \) from \( V \) such that \( f(1) = 0 \).
\[
W = \{ f \in V : f(1) = 0 \}
\]
3. **Question a:**
Prove that space \( W \) is a linear subspace of \( V \).
4. **Question b:**
Let \( D(f(x)) \) be a transformation of \( V \) defined as follows:
\[
D(f(x)) = f'(x)
\]
Prove that \( D \) is a linear transformation. Note that \( D \) is a derivative of \( f \).
5. **Question c:**
Find the matrix of \( D \) in the basis \( \vec{v}_1 = x^3 \), \( \vec{v}_2 = x^2 \), \( \vec{v}_3 = x \), \( \vec{v}_4 = 1 \). Then, find its kernel and range.
---
This problem requires students to demonstrate their understanding of polynomial spaces, subspaces, linear transformations, and matrix representation of these transformations.
**Explanation of Key Concepts:**
1. **Polynomial Spaces:**
A polynomial space of degree 3 consists of all polynomials where the highest degree term is \( x^3 \). For example, \( f(x) = ax^3 + bx^2 + cx + d \).
2. **Subspaces:**
A subspace is a subset of a vector space that itself forms a vector space under the same operations of addition and scalar multiplication.
3. **Linear Transformations:**
A transformation \( D \) is linear if it satisfies additivity and homogeneity:
- Additivity
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