Looking for something else?

Not the answer you are looking for? Search for more explanations.

- anonymous

Let V be the set of all continuous functions defined on [0,1].
Explain why V is a subspace of T(R).
Let W be the subset of V defined by W = {f E V: ∫(0 to 1) (f(t)dt = 0)}. Show that W is a subspace of V.
For the first part, it looks like I need to show:
1) zero vector of T(R) is in V
2) when u and v belong to V, u + v belongs to V
3) when u belongs to V and c is scalar, cu belongs to V
How would I show that the zero function lies in V?
Is it V(0) = 0?
I'm not sure what to do next.

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get our expert's

answer on brainly

SEE EXPERT ANSWER

Get your **free** account and access **expert** answers to this

and **thousands** of other questions.

Get your **free** account and access **expert** answers to this and **thousands** of other questions

- anonymous

- schrodinger

See more answers at brainly.com

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your **free** account and access **expert** answers to this

and **thousands** of other questions

- anonymous

So there are two things that need to be proven here. First, that V is a subspace of T(R) and second that W is a subspace of V. I will show you both below.
1) To prove V is a subspace of T(R), first we need to show that V contains the zero vector. Meaning that for its interval of domain ( D=[0,1]), one of the continuous functions equals zero. And thats easy to show because V contains all continuous functions in that domain D. So just pick a function, maybe f(x)= x, and then we see that f(0)=0. Thats a zero thats contained in V. Next we need to show that the sum of two vectors contained in V are in V and that the product of a scalar k in T(R) and a continuous function in V is in V. Well by definition of continuous functions- the sum of two continuous functions is continuous and the product of a constant with a continuous function is also continuous (http://www.jtaylor1142001.net/calcjat/DEFINITN/ContRules.html).
This finishes part 1.
2) I'm not sure if you needed help with this part, but the same rules apply. The sum of the integrals of two functions is just the integral of those two functions, which is just the integral of another function. And the product of a constant k in T(R) and the integral of a function is just another integral of a function- so there goes that. Oh, and I believe for the zero vector, W already contains it because ∫(0 to 1) (f(t)dt = 0. Ask if you have any more questions.

- anonymous

thanks, that makes sense

Looking for something else?

Not the answer you are looking for? Search for more explanations.