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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
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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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^
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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