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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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Let (X, T) and (Y, T₁) be two topological spaces and let f be a continuous mapping of X into Y.
If f is onto and (Y,T_1) is a T_1 space then
(X,T) is a T_1 space.
If f is one-to-one and (Y,T_1 ) is a T_1 space
then (X,T) is a T_1 space.
None of the choices.
If (Y,T_1) is a Hausdorff space then (X,T) is
a Hausdorff space.

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Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?
Suppose that the second order partial derivatives, fx,y and fy,x, are both continuous on an open set V in R2. Use Fubini’s theorem to prove that fx,y = fy,x in V . Hint: if fx,y(a) − fy,x(a) > 0, there is a rectangle R containing a on which fx,y − fy,x > 0.
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Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)
Let X and Y be topological spaces and endow X×Y and Y×X with the respective product topologies. Show that the map f:X×Y→Y×X defined by f(x,y)=(y,x) is a homeomorphism.
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