Let ?:ℝ^2 → ℝ^2 defined by ?(?) = 1 /2?. Show that f is a contracting mapping.
Let ?:ℝ^2 → ℝ^2 defined by ?(?) = 1 /2?. Show that f is a contracting...
Let V be the set of all functions f : ℝ ℝ discontinuous at each real number, + be the function addition operation, and the multiplication of functions by real constants. What linear space axiom(s) does the structure (V, +, ℝ, ) fail to satisfy?
Let α, β, γ ∈ ℝ designate pairwise different real numbers and understand the ℝ-vectorspace P3(ℝ) of real polynomials of degree 2 or less as an inner product space via. = p(α)q(α) + p(β)q(β) + p(γ)q(γ). Now let λ ∈ C / ℝ designate a complex number which is NOT a real number. Question: Show that for every p, q ∈ P3(ℝ) it holds that is a real number. (Hint: show that the number doesn't change through complex conjugation. (NOTE:...
Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18. Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18.
Let T:ℙ2(ℝ)→ℙ2(ℝ) be a linear transformation given by T(f(x))=3f′(x)+9f(x). If TS:ℝ3→ℝ3 is the corresponding coordinate transformation with respect to the standard basis for P2, {1,x,x2}, compute the matrix AS of the coordinate transformation. (Hint: Consider how T transforms an arbitrary polynomial of the form f(x)=a+bx+cx2.) AS= ⎡⎣⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥
Let X = ℝ with the standard topology and I = [0, 1]. Let F1 be the subset of I formed by removing the open middle third (1/3, 2/3). Then F1 = [0, 1/3]⋃[2/3, 1] Next, let F2 be the subset of F1 formed by removing the open middle thirds (1/9, 2/9) and (7/9, 8/9) of the two components of F1. Then F2 = [0, 1/9] ⋃[2/9, 1/3] ⋃[2/3, 7/9] ⋃[8/9, 1] Continuing this manner, let Fn+1be the subset of...
you can skip #2 Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u where f(r,y,) = =- +22 2. Consider the vector field F(E,) = (a,y) Compute the flow lines for this vector field. 3. Compute the divergence and curl of the following vector field: F(x,y,)(+ yz, ryz, ry + 2) Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u...
h-." 72 16, Let T : R2 → M2×2 be the mapping defined by T ( :' Show that T is a linear transformation.
3. Let f(r) be defined by and let F(x) be defined by F(x) = Í f() dt, a. Find F(x). 0 x 2. For what value of b in the definition of f is F(x) differentiable for all x E [0, 2)?
Let f (2) be defined by: k-?, <<-1 f(3) = z? +, -1<x<1 - kr1 Which of the following values of k would make f (2) continuous on R? Ok=0 There is no such value for k Ok= -1 Ok= 1
Let f:04 →U14 be defined by f(x) = x². Find the kernel off and show that it is a normal subgroup of U14. Show detail step of solution.