3. Consider the set S {1x², -x + 2x2,1 + 2x - x²} in P2 (R). Attempting to find a linear dependence is equivalent to finding a non-trivial solution (a, b, c) to a system of linear equations = a(1-x²) + b(-x+ 2x²) + c(1 + 2x - x²) = =0⇒>> (0 0 0) 11 a+c=0 -b+2c=0 -a +2b-c=0 C:
3. Consider the set S {1x², -x + 2x2,1 + 2x - x²} in P2 (R). Attempting to find a linear dependence is equivalent to finding a non-trivial solution (a, b, c) to a system of linear equations = a(1-x²) + b(-x+ 2x²) + c(1 + 2x - x²) = =0⇒>> (0 0 0) 11 a+c=0 -b+2c=0 -a +2b-c=0 C:
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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Please do it by using matrix. Thank you!
![**Consider the set \( S = \{ 1 - x^2, -x + 2x^2, 1 + 2x - x^2 \} \) in \( P_2(\mathbb{R}) \). Attempting to find a linear dependence is equivalent to finding a non-trivial solution \( (a, b, c) \) to a system of linear equations**
\[
a(1-x^2) + b(-x+2x^2) + c(1+2x-x^2) = 0 \iff
\begin{cases}
a + c = 0 \\
-b + 2c = 0 \\
-a + 2b - c = 0
\end{cases}
\]
Since the only solution is trivial \( (a, b, c) = (0, 0, 0) \), the set \( S \) is linearly independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ada9a20-cfc4-4295-96e5-fd4aa8bf1cec%2Fb66051fd-f495-43f6-8823-4bdca07710e5%2Fzm0sx7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the set \( S = \{ 1 - x^2, -x + 2x^2, 1 + 2x - x^2 \} \) in \( P_2(\mathbb{R}) \). Attempting to find a linear dependence is equivalent to finding a non-trivial solution \( (a, b, c) \) to a system of linear equations**
\[
a(1-x^2) + b(-x+2x^2) + c(1+2x-x^2) = 0 \iff
\begin{cases}
a + c = 0 \\
-b + 2c = 0 \\
-a + 2b - c = 0
\end{cases}
\]
Since the only solution is trivial \( (a, b, c) = (0, 0, 0) \), the set \( S \) is linearly independent.
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