Show that W = {f : R → R : f(1) = 0} together with usual...

Show that W = {f : R → R : f(1) = 0} together with usual addition and scalar multiplication forms a vector space. Let g : R → R and define T : W → W by T(f) = gf. Show that T is a linear transformation.

Expert Solution

Colby Messinger answered 2 years ago

Suppose a function f : R → R is continuous with f(0) = 1. Show that...

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Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R). Explain why {f(x) : f(0) = 1} is not.

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Show that the set of all solutions to the differential equation f′′+f=0, where f∈F(R), is a real vector space with respect to the usual definition of vector addition and scalar multiplication of functions

# O W L S f(O,W,L,S) 0 0 0 0 0 0 1 0 0 0...

# O W L S f(O,W,L,S) 0 0 0 0 0 0 1 0 0 0 1 0 2 0 0 1 0 1 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 1 6 0 1 1 0 1 7 0 1 1 1 X 8 1 0 0 0 0 9 1 0 0 1 0 10 1 0 1 0 0 11 1 0 1 1 1 12 1...

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Which of the following functions are one-to-one? Group of answer choices f;[−3,3]→[0,3],f(x)=9−x2f;[−3,3]→[0,3],f(x)=9−x2 f;R→R,f(x)=x3f;R→R,f(x)=x3 f;[0,∞]→[0,∞],f(x)=x2f;[0,∞]→[0,∞],f(x)=x2

Which of the following functions are one-to-one? Group of answer choices f;[−3,3]→[0,3],f(x)=9−x2f;[−3,3]→[0,3],f(x)=9−x2 f;R→R,f(x)=x3f;R→R,f(x)=x3 f;[0,∞]→[0,∞],f(x)=x2f;[0,∞]→[0,∞],f(x)=x2

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A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty ) f(x)dx = 1. For which value(s) of k e R is the function f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function? Explain.

Let f: [0 1] → R be a function of the class c ^ 2 that...

Let f: [0 1] → R be a function of the class c ^ 2 that satisfies the differential equation f '' (x) = e^xf(x) for all x in (0,1). Show that if x0 is in (0,1) then f can not have a positive local maximum at x0 and can not have a negative local minimum at x0. If f (0) = f (1) = 0, prove that f = 0

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Show that W = {f : R → R : f(1) = 0} together with usual...

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