Clarity on how I should approach this function problem.

[mahjk17]

"If I let V=c([a,b]) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0<=t<=1. What are some conditions that define subspaces of V?" For example will f(1-t) = -tf(t) be a subspace of V?

My attempt:

I assume that it is not a subspace because in order it to be one it will have to satisfy three axioms which are scalar under multiplication, addition, and zero factor. Since it is a continuous interval through 0 <=t<=1 it will not satisfy under addition because it does not span to -f(t).

Is that somewhat right or am I way off? What will be a formal way to define the subspace of this function?

 

[HallsofIvy]

"If I let V=c([a,b]) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0<=t<=1. What are some conditions that define subspaces of V?" For example will f(1-t) = -tf(t) be a subspace of V?

My attempt:

I assume that it is not a subspace because in order it to be one it will have to satisfy three axioms which are scalar under multiplication, addition, and zero factor. Since it is a continuous interval through 0 <=t<=1 it will not satisfy under addition because it does not span to -f(t).

Is that somewhat right or am I way off? What will be a formal way to define the subspace of this function?

Do you understand that you are asserting that the set this problem tells you is a vector space is not a vector space?
The difficulty is that you are not understanding the difference between values of t and f(t). The requirement is that the independent variable, t, between 0 and 1. That has nothing to do with the values of f(t). For example, if f(t)= x+ 2, with domain \(\displaystyle 0\le x\le 1\), then g= 3f is the function with values \(\displaystyle g(x)=3x+ 6\), still with \(\displaystyle 0\le x\le 1\).

When we are given a subset of a vector space, we are using the same definitions of addition and scalar multiplication so properties of those alone, such as commutativity of addition, existance of an additive identity and additive inverses, and the fact that multiplication distributes over addition, are automatic. The only things you need to show for the subset is "closure under addition" and "closure under scalar multiplication". That is, that is u and v are in the subset, so is u+ v and if u is in the subset and a is a scalar (number) then au is also in the set.

For example, the subset of functions such that f(1- t)= -tf(t) is a subspace because if f and g are in this set, f+ g also satisfies that condition- that (f+ g)(1- t)= f(1-t)+ g(1- t)= -tf(t)- tg(t)= -t(f(t)+ g(t)). Also, if f is in this set and a is a number, af also satisfies that condition- that (af)(1- t)= af(1- t)= a(-tf(t))= -t(af(t)).

 

[mahjk17]

I am very great full for this reply, thank you very much! Concise and to the point, I understand functions a bit better now. I have a solid intuition on vector space and subspace but my weakness here was functions but you cleared that up. Thanks again HallsofIvy!