Assume that f:R² → R² is a continuous bijection such that the image of each line is a line. Show that f is affine.
Q: Let F(x, y, z) = y° z°i+ 2xyz³j+ 3xy° z²k. (a) Find the domain of F, and show that F is…
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Q: Show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2).…
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Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…
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Q: Use the sub no spaces: T-2y = - %3D 3x + 2y
A: Given that: x-2y=-10 ........(1)3x+2y=-14 ..............(2)
Q: Let X and Y be two compact spaces, show that X × Y is compact
A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…
Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric
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Q: let (X; Z) v it piece wise comnected ?
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Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…
A: A matric space X is separable if it has a countable dense subset.
Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2
A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…
Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…
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Q: Determine the image of the line Im(z) = -2 under the mapping f(z) = i(7 )? and describe it in words
A: Find the image of the line Imz=-2 under the mapping fz=iz2. Let w=fz, and write w=u+iv, and…
Q: Construct a linear map L(z)=az+b that will send the imaginary y-axis to the line v=-u.
A: we have to constract a linear map that maps the y-axis to the line v=-u Basically we have to…
Q: Determine the image of the line Im (z) = -2 under the mapping f(z)3Di(z) and describe it in words.
A: Given Imz=−2 and a mapping fz=iz¯2 We know that z=x+iy Since, Imz=−2 Hence, y=-2 Now,…
Q: Let R be the parallelogram enclosed by the lines x + 3y = 0, x + 3y = 2, x + y = 1 and x + y = 4.…
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Q: Show that dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2), where W1 + W2 just denotes the span of W1…
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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…
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Q: Prove that linear maps are bounded.
A: To prove that linear maps are bounded.
Q: The mapping T : R² → R defined as T(u) = ||u||
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Q: الموضوع show that if foret A frix by and ; YーZ is a continuous map then gFo relA gf,:X b Z
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Q: Suppose that the three numbers r1, r2, and r3 are dis- tinct. Show that the three functions…
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Q: Determine the image of the line Im(z)= -2 under the mapping f(z)= i(z ) and describe it in words.
A: Im(z)=-2 represents a straight line y=-2let z=x-2i where x is real number. ⇒u+iv=f(z)=iz2=ix+2i2…
Q: Let f : [a, b] → R be a two-to-one map, i.e., for every y E Im(f), there are exactly two points a1,…
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Q: Prove the following relation in the vector triple product : Ax(BxC) + Bx(CxA)+Cx(A x B) = 0
A: The answer is given in handwritten form in the step-2 box. Please check it.
Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…
A: The given problem is related with projection. Given that T is a projection of the vector space V .…
Q: Let Q be an orthogonalmatrix such that QA makes sense. Show that (QA)+ = A+QT.
A: The (Moore-Penrose) pseudoinverse of a matrix generalizes the notion of an inverse,somewhat like the…
Q: Compute the norms for (a)||f(1)||, (b) ||F(x)|l2 (c)||f(x)||, where f(r) = -VI-r, defined on [0, 1).
A: Given- fx =-1-x defined on [0, 1] (a) we have to calculate fx fx=∫01fxdx =∫01-1-xdx…
Q: Let T : R3 → R² be a linear mapping. 2 .T Given that T and T find T
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Q: Show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1,…
A: For satisfying the inner product condition, function has to follow these three conditions: 1)…
Q: im(T1 + 14T) im(14T† + T2).
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Q: Let f(x) = V-36 - z and g(x) = a - 13z. fog= %3D The domain offogis:
A: Recall: A composite function is created when one function is substituted into another function.…
Q: The mapping w = z² + 1 is not conf ormal at O z = 0 O z = i O z = -i
A: We know that analytical function is not conformal at critical points. Let's find the critical points…
Q: b.) answer. Let f = x² and g = (1-x)². Are x and y orthogonal in this inner product space? Justify…
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Q: 3. (a) Show that under the mapping w = 1/z, all circles and straight lines in the =-plane are…
A: (a) We have to show under the mapping w = 1/z, all circles and straight lines in the z plane are…
Q: Show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1,…
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Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…
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Q: Q/ if f contineous map Prove that f(B) ≤ f(B), VBsy
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Q: Let w = f(u), where u = x g(y), show that w,xx = f" (x g(y)). g²(y). %3D
A: Let w=f(u), where u=x g(y). So using the chain rule, differentiating with respect to x, we get…
Q: Let F(x, y) = <3x?y3 + y", 3x°y² + y^ + 4xy³,. %3D Evaluate F. dr where C is the line segment from…
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Q: Show that f : R → R is bijective, where: f(x) %3D필
A: A function is injective if f(x)=f(y)⇒x=y. A function is onto if for every y in Y there exists a x in…
Q: Consider the space L'[0,1]. Let p(f) :
A: Given, pf=∫1234f212, f∈L20,1. Standard L2-norm is f2=∫01f212
Q: Find the image of | z – 3i | = 3 under the mapping W =-.
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Q: State whether the following statement is true or false: If f:U V is a linear map, then Kerf + Ø O…
A: kernel of a function is the set of all elements which map to 0.
Q: Sketch the set of all points (x, y) such that x + y| < e*.
A: Consider the function x+y≤ex. An absolute function is defined as x=x if x≥0-x if x<0. So, there…
Q: the euclidean space R³ is a separable metric space. true or false?
A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…
Q: Show that X is Hausdorff if and only the diagonal A = {(x,x) | x € X} is closed in X x X.
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Q: Describe geometrically the set of points z satisfying the following condition:|z−2i|=3 .
A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…
Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).
A: Since you have asked multiple question, we will solve the first question for you. If you want any…
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- If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
- Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy
- Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function defined on R, such that for every x it holds that |a(x)| ≤ 1. Using the Global Picard–Lindel¨of Theorem, show that there exists a unique solution y defined on R.Integrate the function f(x, y, z) = 0.7(x2 + y2 + z2 ) over the unit sphere S ={(x,y, z) | x2 +y2+z2 ≤ 1} using the Monte-Carlo method in three dimensions. using a sample of M = 106 points.If E is Lebesgue measurable subset of [a, b], show thatZ baχE = m(E)by using the definition of the Lebesgue integral.
- Show that if f(x) > 0 for all x ∈ [a, b] and f is integrable, then ) b a f > 0.1 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.Consider the function f(x) = ln(x)/x^5. f(x) has a critical number A = __? f"(A) = __? Thus we conclude that f(x) has a local __ at A (type in MAX or MIN).