= 5.8. Let f₁(x) = x³ x² +1, f₂(x) = x³ + x² + x and f3 (x) = x³ + 2x²+3x+1 be three polynomials in the vector space P3 and let W Span{f(x), f₂(x), ƒ3 (x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), f(x).
= 5.8. Let f₁(x) = x³ x² +1, f₂(x) = x³ + x² + x and f3 (x) = x³ + 2x²+3x+1 be three polynomials in the vector space P3 and let W Span{f(x), f₂(x), ƒ3 (x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), f(x).
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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![### Polynomial Subspaces and Linear Combinations
#### Problem Statement:
Let \( f_1(x) = x^3 - x^2 + 1 \), \( f_2(x) = x^3 + x^2 + x \), and \( f_3(x) = x^3 + 2x^2 + 3x + 1 \) be three polynomials in the vector space \( P_3 \), and let \( W = \text{Span}\{f_1(x), f_2(x), f_3(x)\} \).
Does the polynomial \( g(x) = 4x^3 + 11x^2 + 11x \) belong to the subspace \( W \)? If yes, write \( g(x) \) as a linear combination of the vectors \( f_1(x), f_2(x), f_3(x) \).
### Solution:
To determine if \( g(x) \) belongs to the subspace \( W \), we need to express \( g(x) \) as a linear combination of \( f_1(x), f_2(x), f_3(x) \). In other words, we need to find scalars \( a \), \( b \), and \( c \) such that:
\[ g(x) = a \cdot f_1(x) + b \cdot f_2(x) + c \cdot f_3(x) \]
where:
\[ f_1(x) = x^3 - x^2 + 1 \]
\[ f_2(x) = x^3 + x^2 + x \]
\[ f_3(x) = x^3 + 2x^2 + 3x + 1 \]
\[ g(x) = 4x^3 + 11x^2 + 11x \]
### Steps to Solve:
1. **Write the linear combination:**
\[ a(x^3 - x^2 + 1) + b(x^3 + x^2 + x) + c(x^3 + 2x^2 + 3x + 1) \]
2. **Combine like terms:**
\[ (a + b + c)x^3 + (-a + b + 2c)x^2 + (b + 3c)x + a + c = 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4663db8c-9c2f-44c5-826e-200804360ec0%2F6c6c662b-d7bf-45f1-9a37-bf9af25d12fe%2F7uucwqs_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Subspaces and Linear Combinations
#### Problem Statement:
Let \( f_1(x) = x^3 - x^2 + 1 \), \( f_2(x) = x^3 + x^2 + x \), and \( f_3(x) = x^3 + 2x^2 + 3x + 1 \) be three polynomials in the vector space \( P_3 \), and let \( W = \text{Span}\{f_1(x), f_2(x), f_3(x)\} \).
Does the polynomial \( g(x) = 4x^3 + 11x^2 + 11x \) belong to the subspace \( W \)? If yes, write \( g(x) \) as a linear combination of the vectors \( f_1(x), f_2(x), f_3(x) \).
### Solution:
To determine if \( g(x) \) belongs to the subspace \( W \), we need to express \( g(x) \) as a linear combination of \( f_1(x), f_2(x), f_3(x) \). In other words, we need to find scalars \( a \), \( b \), and \( c \) such that:
\[ g(x) = a \cdot f_1(x) + b \cdot f_2(x) + c \cdot f_3(x) \]
where:
\[ f_1(x) = x^3 - x^2 + 1 \]
\[ f_2(x) = x^3 + x^2 + x \]
\[ f_3(x) = x^3 + 2x^2 + 3x + 1 \]
\[ g(x) = 4x^3 + 11x^2 + 11x \]
### Steps to Solve:
1. **Write the linear combination:**
\[ a(x^3 - x^2 + 1) + b(x^3 + x^2 + x) + c(x^3 + 2x^2 + 3x + 1) \]
2. **Combine like terms:**
\[ (a + b + c)x^3 + (-a + b + 2c)x^2 + (b + 3c)x + a + c = 4
![### Linear Independence and Function Decomposition
**Example 5.8:**
Let \( g(x) = c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) \), for all \( x \). Simplify this to get:
\[ (c_1 + c_2 + c_3 - 4)x^3 + (-c_1 + c_2 + 2c_3 - 11)x^2 + (c_2 + 3c_3 - 11)x + c_1 + c_3 = 0, \]
for all \( x \).
Since \( 1, x, x^2, x^3 \) are linearly independent, we must have:
\[
\begin{cases}
c_1 + c_2 + c_3 - 4 = 0, \\
-c_1 + c_2 + 2c_3 - 11 = 0, \\
c_2 + 3c_3 - 11 = 0, \\
c_1 + c_3 = 0.
\end{cases}
\]
Solving these equations, we get:
\[
c_1 = -\frac{7}{3}, \quad c_2 = 4, \quad c_3 = \frac{7}{3}.
\]
Hence, \( g(x) \) is in \( W \) and
\[ g(x) = -\frac{7}{3}f_1(x) + 4f_2(x) + \frac{7}{3}f_3(x). \]
### Explanation of the Equations:
- The problem begins by expressing \( g(x) \) as a linear combination of functions \( f_1(x), f_2(x), \) and \( f_3(x) \).
- The resulting equation is simplified and represented in a polynomial form to consider the coefficients of \( x^3, x^2, x \), and the constant term.
- Since the polynomials are linearly independent, their coefficients must equal zero.
### Solution Process Breakdown:
1. Write \( g(x) \) as a linear combination:
\[ g(x) = c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4663db8c-9c2f-44c5-826e-200804360ec0%2F6c6c662b-d7bf-45f1-9a37-bf9af25d12fe%2Fao2zj1h_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Independence and Function Decomposition
**Example 5.8:**
Let \( g(x) = c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) \), for all \( x \). Simplify this to get:
\[ (c_1 + c_2 + c_3 - 4)x^3 + (-c_1 + c_2 + 2c_3 - 11)x^2 + (c_2 + 3c_3 - 11)x + c_1 + c_3 = 0, \]
for all \( x \).
Since \( 1, x, x^2, x^3 \) are linearly independent, we must have:
\[
\begin{cases}
c_1 + c_2 + c_3 - 4 = 0, \\
-c_1 + c_2 + 2c_3 - 11 = 0, \\
c_2 + 3c_3 - 11 = 0, \\
c_1 + c_3 = 0.
\end{cases}
\]
Solving these equations, we get:
\[
c_1 = -\frac{7}{3}, \quad c_2 = 4, \quad c_3 = \frac{7}{3}.
\]
Hence, \( g(x) \) is in \( W \) and
\[ g(x) = -\frac{7}{3}f_1(x) + 4f_2(x) + \frac{7}{3}f_3(x). \]
### Explanation of the Equations:
- The problem begins by expressing \( g(x) \) as a linear combination of functions \( f_1(x), f_2(x), \) and \( f_3(x) \).
- The resulting equation is simplified and represented in a polynomial form to consider the coefficients of \( x^3, x^2, x \), and the constant term.
- Since the polynomials are linearly independent, their coefficients must equal zero.
### Solution Process Breakdown:
1. Write \( g(x) \) as a linear combination:
\[ g(x) = c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) \]
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