skoomafiend said:Homework Statement
V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.
is V a vector space given the operations?
Homework Equations
nil.
The Attempt at a Solution
i think it is not closed under multiplication.
if r is an element of R, then
r*a(x) . r*f(x) = (r^2)*(a.f)(x)
which is not equal to
r*f(ax)
im not really sure if i even have the correct approach.
any help would be greatly appreciated.
thanks!
skoomafiend said:Homework Statement
V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.
If I'm understanding the problem correctly, f(x) = x2 is not a member of set V, since af(x) [itex]\neq[/itex] f(ax).Deveno said:suppose that f(x) = x2.
is it the case that ((a+b).f)(x) = (a.f)(x) + (b.f)(x)?
(this is the distributivity of field addition over scalar mutliplication axiom).
Mark44 said:If I'm understanding the problem correctly, f(x) = x2 is not a member of set V, since af(x) [itex]\neq[/itex] f(ax).
A vector space of functions is a mathematical structure that consists of a set of functions and two operations, addition and scalar multiplication, that satisfy certain properties. These properties include closure under addition and scalar multiplication, associativity, commutativity, and the existence of an identity element and inverse element.
The main difference between vector spaces of functions and regular vector spaces is the type of elements they contain. Regular vector spaces contain vectors, which are defined as a combination of numbers, while vector spaces of functions contain functions as their elements. Additionally, vector spaces of functions have operations defined on them, such as addition and scalar multiplication, that are specific to functions.
Some common examples of vector spaces of functions include the space of all polynomials of a certain degree, the space of all continuous functions on a given interval, and the space of all differentiable functions on a given interval. These spaces are often used in mathematical analysis and are essential in many areas of mathematics and physics.
The dimension of a vector space of functions is the number of linearly independent functions that span the space. This is similar to the dimension of a regular vector space, which is the number of linearly independent vectors that span the space. However, in a vector space of functions, the basis elements are functions rather than vectors.
Vector spaces of functions have many practical applications in fields such as engineering, physics, and computer science. For example, they are used in signal processing to analyze and manipulate signals, in control theory to model and control systems, and in computer graphics to represent and manipulate images and shapes. They are also heavily used in mathematical physics, where they provide a powerful framework for representing and solving physical problems.