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1)Complete each of the following statements using the words “greater than”, “less than” or “equal to” a) The cardinality of the even numbers is _________________ the cardinality of the natural numbers. b) The cardinality of the natural numbers is _________________ the cardinality of the positive rational numbers. c) The cardinality of the natural numbers is _________________ the cardinality of the rational numbers. d) The cardinality of the real numbers is _________________ the cardinality of the natural numbers. e) The cardinality...
Define two functions T: ℝ^2 ⟶ ℝ^3 and S: ℝ^2 ⟶ ℝ^2 by T [X, Y] = [2x + y, 0], S[x,y] = - [x +y, xy] Determine whether T and S, and the composite S ∘ T are linear transformations.
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 ?:ℝ^2 → ℝ^2 defined by ?(?) = 1 /2?. Show that f is a contracting mapping.
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 A∈ℝnxn, and suppose ?,?∈ℝ with ?≠0 such that ?+?i is an eigenvalue of ?. Suppose vectors ?,?∈ℝ? such that ?+?? is an eigenvector for ? associated with the eigenvalue ?+?i. Prove that vectors ? and ? are linearly independent.
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...
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= ⎡⎣⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥
Is T: M2,2 → ℝ defined by T(A) =|A| a linear transformation? Provide a proof or counterexample.
Use Cantor Diagonal Argument to prove that the set {?∈ℝ|9≤?<10} is uncountable infinite.