19. r₁ = 2 - 3x + x², 1₂ 20. s₁ = 1 + x², S₂ = x + x³, S3 = x + x², 54 = 1 + x³, for P₁ le 1 and find a basis for M32. What is dim M
19. r₁ = 2 - 3x + x², 1₂ 20. s₁ = 1 + x², S₂ = x + x³, S3 = x + x², 54 = 1 + x³, for P₁ le 1 and find a basis for M32. What is dim M
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Do the given vectors form a basis for the given vector space?
(Number 20 on picture)
![## Vector Spaces
### Exercises:
19. \( r_1 = 2 - 3x + x^2, \, r_2 = 4 + 3x - 2x^2, \, r_3 = 9x - 4x^2 \), for \( P_2 \).
20. \( s_1 = 1 + x + x^2, \, s_2 = x + x^3, \, s_3 = x + x^2, \, s_4 = 1 + x^3 \), for \( P_3 \).
21. Follow the pattern of Example 4 and find a basis for \( M_{32} \). What is \(\dim M_{32}\)?
22. Same question as Exercise 21 for \( M_{22} \).
23. Same question as Exercise 21 for \( M_{mn} \).
24. Determine if the matrices
\[
M_1 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad
M_2 = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \quad
M_3 = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \quad
M_4 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}
\]
form a basis for \( M_{22} \).
25. Same question as Exercise 24 for
\[
N_1 = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, \quad
N_2 = \begin{bmatrix} -2 & 3 \\ 2 & 0 \end{bmatrix}, \quad
N_3 = \begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}, \quad
N_4 = \begin{bmatrix} 0 & 1 \\ 0 & 4 \end{bmatrix}
\]
26. Let \( W \) be the subspace of \( C[0, 1] \) spanned by \( S = \{\sin^2 x, \cos^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92dd192b-2b85-4e23-91c3-a8e23aec13bb%2F8f4eed22-7c97-41d7-816a-22ddf83491ff%2Fj9lec8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Vector Spaces
### Exercises:
19. \( r_1 = 2 - 3x + x^2, \, r_2 = 4 + 3x - 2x^2, \, r_3 = 9x - 4x^2 \), for \( P_2 \).
20. \( s_1 = 1 + x + x^2, \, s_2 = x + x^3, \, s_3 = x + x^2, \, s_4 = 1 + x^3 \), for \( P_3 \).
21. Follow the pattern of Example 4 and find a basis for \( M_{32} \). What is \(\dim M_{32}\)?
22. Same question as Exercise 21 for \( M_{22} \).
23. Same question as Exercise 21 for \( M_{mn} \).
24. Determine if the matrices
\[
M_1 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad
M_2 = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \quad
M_3 = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \quad
M_4 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}
\]
form a basis for \( M_{22} \).
25. Same question as Exercise 24 for
\[
N_1 = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, \quad
N_2 = \begin{bmatrix} -2 & 3 \\ 2 & 0 \end{bmatrix}, \quad
N_3 = \begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}, \quad
N_4 = \begin{bmatrix} 0 & 1 \\ 0 & 4 \end{bmatrix}
\]
26. Let \( W \) be the subspace of \( C[0, 1] \) spanned by \( S = \{\sin^2 x, \cos^
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