liñe if are subspaces of F[a, b]. a) All functions ƒ in F[a, b] for which f (a) = 0 b) All functions f in F[a, b] for which f (a) = 1 c) All functions f in C[a, b] for which S. f(x) dx = 0 d) All...

Can you do number 9 show all work ? Vector space


Extracted text: liñe if are subspaces of F[a, b]. a) All functions ƒ in F[a, b] for which f (a) = 0 b) All functions f in F[a, b] for which f (a) = 1 c) All functions f in C[a, b] for which S. f(x) dx = 0 d) All functions f in D[a, b] for which f'(x) = f(x) e) All functions f in D[a, b] for which f'(x) = e* 1 16. Determine if 1 1 - 17. Determine if 4. Determine which of the following sets of n × n ma- trices are subspaces of Mnxn (IR). 1 1 a) The n x n diagonal matrices b) The n x n upper triangular matrices c) The n x n symmetric matrices 1 18. Determine if d) The n x n matrices of determinant zero e) The n x n invertible matrices 5. If A is an m x n matrix and B of R", do the solutions to the system AX = B form a subspace of R"? Why or why not? 6. Complex numbers a + bi where a and b are integers are called Gaussian integers. Do the Gaussian inte- gers form a subspace of the vector space of complex numbers? Why or why not? a nonzero element 19. Determine if x² – 20. Determine if x³ +. Use the system of linea Maple or another apprc cises 21-24. 7. Do the sequences that converge to zero form a sub- space of the vector space of convergent sequences? How about the sequences that converge to a rational number? 8. Do the series that converge to a positive number form a subspace of the vector space of convergent series? How about the series that converge absolutely? 21. Determine if [] 9. Is in Span 4 -4 10. Is -2 in Span 3 -3 1 1 Span 2 11. Is -5 in Span 3 -3 1 3X L 372 L3X, 4312