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Exercise 6.3.6 Exhibit a basis and calculate the dimen- sion of each of the following subspaces of P2. a. {a(1+x)+b(x+x²) | a and b in R} b. {a+b(x+x²) | a and b in R} c. {p(x) | p(1)=0} d. {p(x)|p(x) = p(−x)}

Exercise 6.3.6 Exhibit a basis and calculate the dimen- sion of each of the following subspaces of P2. a. {a(1+x)+b(x+x²) | a and b in R} b. {a+b(x+x²) | a and b in R} c. {p(x) | p(1)=0} d. {p(x)|p(x) = p(−x)}

Advanced Engineering Mathematics
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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c and d only
**Exercise 6.3.6** Exhibit a basis and calculate the dimension of each of the following subspaces of \( P_2 \). a. \( \{a(1+x) + b(x+x^2) \mid a \text{ and } b \in \mathbb{R}\} \) b. \( \{a + b(x+x^2) \mid a \text{ and } b \in \mathbb{R}\} \) c. \( \{p(x) \mid p(1) = 0\} \) d. \( \{p(x) \mid p(x) = p(-x)\} \) **Exercise 6.3.7** Exhibit a basis and calculate the dimension of each of the following subspaces of \( M_{22} \). a. \( \{A \mid A^T = -A\} \) b. \( \left\{ A \mid A \begin{bmatrix} 1 & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ -1 & 0 \end{bmatrix} A \right\} \)
Transcribed Image Text:**Exercise 6.3.6** Exhibit a basis and calculate the dimension of each of the following subspaces of \( P_2 \). a. \( \{a(1+x) + b(x+x^2) \mid a \text{ and } b \in \mathbb{R}\} \) b. \( \{a + b(x+x^2) \mid a \text{ and } b \in \mathbb{R}\} \) c. \( \{p(x) \mid p(1) = 0\} \) d. \( \{p(x) \mid p(x) = p(-x)\} \) **Exercise 6.3.7** Exhibit a basis and calculate the dimension of each of the following subspaces of \( M_{22} \). a. \( \{A \mid A^T = -A\} \) b. \( \left\{ A \mid A \begin{bmatrix} 1 & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ -1 & 0 \end{bmatrix} A \right\} \)
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