Topic is Linear Algebra. Please help solve. Thank you!! Exercise 1: Let X be any set and consider the set of all functions S = {f: X → R}. Notice we can add two functions(f+g)(x) = f(x) + g(x), we can scale a function (a · ƒ)(x) = a · ƒ(x), and we have a zero function z(x) = 0. Prove twoproperties to start to see that S is a vector space. Now from here...(a) Look at some Calculus textbook (tell me which book you used, the library has some, and the internet has tons :) )and tell me where you found the properties which would help you see why the set of continuous functions with thesame domain is a vector space.(b) Now assuming continuous functions are a vector space, prove to me that differentiable functions are a subspace usingthe definition of subspace and some properties from Calculus.(c) Now assuming differentiable functions are a vector space, prove to me that polynomial functions are a subspace ofdifferentiable functions using the definition of subspace and properties from Algebra.

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Answered: Exercise 1: Let X be any set and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Topic is Linear Algebra. Please help solve. Thank you!!

Exercise 1: Let X be any set and consider the set of all functions S = {f: X → R}. Notice we can add two functions
(f+g)(x) = f(x) + g(x), we can scale a function (a · ƒ)(x) = a · ƒ(x), and we have a zero function z(x) = 0. Prove two
properties to start to see that S is a vector space. Now from here...
(a) Look at some Calculus textbook (tell me which book you used, the library has some, and the internet has tons :) )
and tell me where you found the properties which would help you see why the set of continuous functions with the
same domain is a vector space.
(b) Now assuming continuous functions are a vector space, prove to me that differentiable functions are a subspace using
the definition of subspace and some properties from Calculus.
(c) Now assuming differentiable functions are a vector space, prove to me that polynomial functions are a subspace of
differentiable functions using the definition of subspace and properties from Algebra.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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