(Linear Algebra) For the set below, determine whether it is a vector space (with
ID: 3197010 • Letter: #
Question
(Linear Algebra) For the set below, determine whether it is a vector space (with the defined addition and scalar multiplication). If the set is not a vector space, show that at least one of the properties defining a vector space does not hold:The set of real-valued functions f with f(1)=0; addition is defined as (f+g)(x)=f(x)+g(x); scalar multiplication is defined as (cf)(x)=cf(x) (Linear Algebra) For the set below, determine whether it is a vector space (with the defined addition and scalar multiplication). If the set is not a vector space, show that at least one of the properties defining a vector space does not hold:
The set of real-valued functions f with f(1)=0; addition is defined as (f+g)(x)=f(x)+g(x); scalar multiplication is defined as (cf)(x)=cf(x)
The set of real-valued functions f with f(1)=0; addition is defined as (f+g)(x)=f(x)+g(x); scalar multiplication is defined as (cf)(x)=cf(x)
Explanation / Answer
To prove given set is a vector space under given operation
1) closure under addition
f+g(1) = f(1)+g(1)=0+0=0
Real valued function are clearly commutative, associative, distributive under both addition and multiplication
Only thing left to prove is cf belongs to this set
CF(1) = 0 so it is a vector space
Another easy approach is clearly set of real valued function is a vector space
F=0 is clearly belongs to this set
So if we prove cf+g belongs to this set and
(CF+g)(1)= c*0+0=0
Hence proved
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