Recent content by redyelloworange

for both parts?

Homework Statement Two functions f,g:R->R are equal up to nth order at if lim h-->0 [f(a+h) - g(a+h) / hn ]= 0 Show that f is differentiable at a iff there is a function g(x)=b+m(x-a) such that f and g are equal up to first order at a. Homework Equations f is differentiable at a if...

What's N_e?

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

bd(S) = {x in Rn s.t. B(r,x)∩ S ≠ ø and B(r,x) ∩ Sc≠ ø for every r>0}

Homework Statement Let U and V be open sets in Rn and let f be a one-to-one mapping from U onto V (so that there is an inverse mapping f-1). Suppose that f and f-1 are both continuous. Show that for any set S whose closure is contained in U we have f(bd(S)) = bd(f(S)). Homework Equations...

Homework Statement Prove that every nonempty proper subset of Rn has a nonempty boundry. The Attempt at a Solution First of all, I let S be an nonempty subset of Rn and S does not equal Rn. I tried to go about this in 2 different ways: 1) let x be in S and show that B(r,x) ∩ S ≠ ø and...

<T*(T-1)*x,y> = <T*(x), T-1(y)> = <(T-1)*x,T(y)> = <x, (T-1)(T)y> = <x,y> so T*(T-1)* = I, so (T-1)* is the inverse of T*, hence (T-1)* = (T*)-1) thanks! =)

Sorry about that, It should be T = T_1 + iT_2 T_2 is self adoint: T_2* = (-1/2i)(T*-T) = (1/2i)(T-T*) = T_2

Homework Statement Prove that any square triangular matrix with each diagonal entry equal to zero is nilpotent The Attempt at a Solution Drawing out the matrix and multiplying seems a little tedious. Perhaps there is a better way? Is there another way to do this without assuming that the...

Homework Statement Let T:V  W be a linear transformation. Prove the following results. (a) N(T) = N(-T) (b) N(T^k) = N((-T)^k) (c) If V = W and t is an eigenvalue of T, then for any positive integer k N((T-tI)^k) = N((tI-T)^k) where I is the identity transformation The Attempt at a...

Homework Statement Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) = W (b) the orthogonal complement of W is U-invariant (for ease of writing let the orthogonal complement of W be represented by...

Homework Statement Let V be a complex inner product space, and let T be a linear operator on V. Define T_1 = 1/2 (T + T*) and T_2 = (1/2i)(T – T*) a) Prove that T_1 and T_2 are self-adjoint and that T = T_1 + T_2 b) Suppose also that T = U_1 + U_2, where U_1 and U_2 are self-adjoint...

Homework Statement Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that if T is invertible, then T* is invertible and (T*)^-1 = (T^-1)* Homework Equations As shown above. <T(x),y> = <x,T*(y)> The Attempt at a Solution Well, I figure you...

Homework Statement Let T be a linear operator on a vector space V, and suppose that V is a T-cyclic subspace of itself. Prove that if U is a linear operator on V, then UT=TU if and only if U=g(T) for some polynomial g(t). Homework Equations They suggest supposing that V is generated by...