Q4. For general linear transformations f : R is invertible and g-fo f, prove that g is invertible if f [4 marks Q5. For general inear transformations f:RR and g - fo f, prove that f is invertible...
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r, θ) (i) (7 marks] if f (θ)-sin|-θ | , find the specific solution u (r,0) (ii) [ (ii) [5 marks] Use the solution in (ii) to deduce that 4n1-9) 18 Q4. [24marks] Consider the second order linear partial differential equation Q3. [22 marks] The Dirichlet's...
4. (3 marks) Suppose that you have two linear functions f(x) and g(z) such that f(a) = g(a) = 0. Suppose also that f(k) = 6 and g(k) 3 for some ke R (see figure below). Determine f(x) 6 3
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
Please prove Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(a)-c and f(b)-d and show this is a homeomorphism.) Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function...
Do A and used C as question say A. (This problem gives an explanation for the isomorphism R 1m(A) R"/1m(A'), where A, Q-IAP, with Q and P invertible.) Let R be a ring and let M, N, U, V be R-modules such that there existR module homomorphisms α : M N, β : u--w, γ: M-+ U and δ: N V such that the following diagram is commutative: (recall that commutativity of the diagram means that δ ο α γ)...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
4. Let TB:R" + R" and TA: RM → RP be the linear transformations represented by an mxn matrix B = [bj] and p x m matrix A = [ai;], that is, b11x1 + ... binin y = TB(x): bm121 + ... + bmnan 01191 +... 01mYm 21 :| = z=TA(y) : apıyı + ... + Apmym where x= y= 53 z= : represent vectors in R", RM, RP, respectively. Then, we know from Problem 3 that the com- position...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...