Numerical Analysis, Books A La Carte Edition (3rd Edition)
3rd Edition
ISBN: 9780134697338
Author: Timothy Sauer
Publisher: PEARSON
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Chapter 8.1, Problem 6E
Show that the nonzero
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please answer the questions below ands provide the required codes in PYTHON. alsp provide explanation of how the codes were executed. Also make sure you provide codes that will be able to run even with different parameters as long as the output will be the same with any parameters given. these questions are not graded. provide accurate codes please
(1) Let F be a field, show that the vector space F,NEZ* be a finite dimension.
(2) Let P2(x) be the vector space of polynomial of degree equal or less than two
and M={a+bx+cx²/a,b,cЄ R,a+b=c),show that whether Mis hyperspace or not.
(3) Let A and B be a subset of a vector space such that ACB, show that whether:
(a) if A is convex then B is convex or not. (b) if B is convex then A is convex or not.
(4) Let R be a field of real numbers and X=R, X is a vector space over R show that by
definition the norms/II.II, and II.112 on X are equivalent where
Ilxll₁ = max(lx,l, i=1,2,...,n) and llxll₂=(x²).
oper
(5) Let Ⓡ be a field of real numbers, Ⓡis a normed space under usual operations and
norm, let E=(2,5,8), find int(E), b(E) and D(E).
(6) Write the definition of bounded linear function between two normed spaces and
write with prove the relation between continuous and bounded linear function
between two normed spaces.
ind
→ 6
Q₁/(a) Let R be a field of real numbers and X-P(x)=(a+bx+cx²+dx/ a,b,c,dER},X is
a vector space over R, show that is finite dimension.
(b) Let be a bijective linear function from a finite dimension vector ✓ into
a space Yand Sbe a basis for X, show that whether f(S) basis for or not.
(c) Let be a vector space over a field F and A,B)affine subsets of X,show that
whether aAn BB, aAU BB be affine subsets of X or not, a,ẞ EF.
(12
Jal (answer only two) (6) Let M be a non-empty subset of a vector space X and tEX,
show that M is a hyperspace of X iff t+M is a hyperplane of X and tЄt+M.
(b) State Jahn-Banach theorem and write with prove an application of Hahn-
Chapter 8 Solutions
Numerical Analysis, Books A La Carte Edition (3rd Edition)
Ch. 8.1 - Prove that the functions (a) u(x,t)=e2t+x+e2tx,...Ch. 8.1 - Prove that the functions (a) u(x,t)=etsinx, (b)...Ch. 8.1 - Prove that if f(x) is a degree 3 polynomial, then...Ch. 8.1 - Prob. 4ECh. 8.1 - Verify the eigenvector equation (8.13).Ch. 8.1 - Show that the nonzero vectors vj in (8.12 ), for...Ch. 8.1 - Prob. 1CPCh. 8.1 - Consider the equation ut=uxx for 0x1, 0t1 with the...Ch. 8.1 - Prob. 3CPCh. 8.1 - Use the Backward Difference Method to solve the...
Ch. 8.1 - Use the Crank-Nicolson Method to solve the...Ch. 8.1 - Prob. 6CPCh. 8.1 - Prob. 7CPCh. 8.1 - Setting C=D=1 in the population model (8.26), use...Ch. 8.2 - Prove that the functions (a) u(x,t)=sinxcos4t, (b)...Ch. 8.2 - Prove that the functions (a) u(x,t)=sinxsin2t, (b)...Ch. 8.2 - Prove that u1(x,t)=sinxcosct and u2(x,t)=ex+ct are...Ch. 8.2 - Prove that if s(X) is twice differentiable, then...Ch. 8.2 - Prove that the eigenvalues of A in (8.33) lie...Ch. 8.2 - Let be a complex number. (a) Prove that if +1/ is...Ch. 8.2 - Solve the initial-boundary value problems in...Ch. 8.2 - Solve the initial-boundary value problems in...Ch. 8.2 - Prob. 3CPCh. 8.2 - Prob. 4CPCh. 8.3 - Show that u(x,y)=ln(x2+y2) is a solution to the...Ch. 8.3 - Prob. 2ECh. 8.3 - Prove that the functions (a) u(x,y)=eysinx, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=exy, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=sin2xy, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=ex+2y, (b)...Ch. 8.3 - Prob. 7ECh. 8.3 - Show that the barycenter of a triangle with...Ch. 8.3 - Prove Lemma 8.9 .Ch. 8.3 - Prove Lemma 8.10.Ch. 8.3 - Prob. 11ECh. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Solve the Laplace equation problems in Exercise 3...Ch. 8.3 - Prob. 2CPCh. 8.3 - Prob. 3CPCh. 8.3 - Prob. 4CPCh. 8.3 - Prob. 5CPCh. 8.3 - The steady-state temperature u on a heated copper...Ch. 8.3 - Prob. 7CPCh. 8.3 - Prob. 8CPCh. 8.3 - Solve the Laplace equation problems in Exercise 3...Ch. 8.3 - Solve the Poisson equation problems in Exercise 4...Ch. 8.3 - Solve the elliptic partial differential equations...Ch. 8.3 - Prob. 12CPCh. 8.3 - Prob. 13CPCh. 8.3 - Solve the elliptic partial differential equations...Ch. 8.3 - Prob. 15CPCh. 8.3 - Prob. 16CPCh. 8.3 - For the elliptic equations in Exercise 7, make a...Ch. 8.3 - Solve the Laplace equation with Dirichlet boundary...Ch. 8.4 - Show that for any constant c, the function...Ch. 8.4 - Show that over an interval [ x1,xr ] not...Ch. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 1CPCh. 8.4 - Prob. 2CPCh. 8.4 - Solve Fishers equation (8.69) with...Ch. 8.4 - Prob. 4CPCh. 8.4 - Solve the Brusselator equations for...Ch. 8.4 - Prob. 6CP
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