Let B = {1+t,1 – t} and C = {4, 1 + 5t, t²}. Does B form a basis for P2?
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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![Let \( B = \{ 1 + t^2, 1 - t \} \) and \( C = \{ 4, 1 + 5t, t^2 \} \).
Does \( B \) form a basis for \( P_2 \)?
- \( \circ \) Yes, because it has powers of \( t \) from 0 to 2.
- \( \circ \) No, because its polynomials are not the standard polynomials \( 1, \, t, \) and \( t^2 \).
- \( \circ \) No, because there are not enough vectors in \( B \) to form a basis.
- \( \circ \) Yes, because it contains two linearly independent polynomials and \( P_2 \) is two dimensional.
If \( q = 26 + 10t + 6t^2 \), find \([q]_C\).
\[
\begin{bmatrix}
\\
\\
\\
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5025798a-663c-4f82-a82a-d0677c58328a%2Fa1e899bc-0198-48ea-ab13-fb0723b394f2%2Fxngqg7s_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( B = \{ 1 + t^2, 1 - t \} \) and \( C = \{ 4, 1 + 5t, t^2 \} \).
Does \( B \) form a basis for \( P_2 \)?
- \( \circ \) Yes, because it has powers of \( t \) from 0 to 2.
- \( \circ \) No, because its polynomials are not the standard polynomials \( 1, \, t, \) and \( t^2 \).
- \( \circ \) No, because there are not enough vectors in \( B \) to form a basis.
- \( \circ \) Yes, because it contains two linearly independent polynomials and \( P_2 \) is two dimensional.
If \( q = 26 + 10t + 6t^2 \), find \([q]_C\).
\[
\begin{bmatrix}
\\
\\
\\
\end{bmatrix}
\]
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