Practice question · True or false
The set of all real-valued functions f with f(2) = 0 is a vector space under pointwise addition and scaling.
Hints
- Add two such functions and evaluate the sum at 2.
- Scale one of them by c and evaluate at 2.
Show the answer
True
Why
True. If f(2) = 0 and g(2) = 0 then (f + g)(2) = 0 and (cf)(2) = 0, so the set is closed; the zero function belongs to it, and the remaining axioms are inherited from the space of all functions. Had the condition been f(2) = 1, closure would fail immediately.
Practise Vector Spaces
The app has 5 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Vector Spaces
- Polynomials of degree at most 2, 2x3 matrices, and arrows in the plane are studied as one subject. Why is it…
- The solutions of a linear differential equation form a vector space. Why is noticing that worth anything to…
- Select every set that IS a vector space under the usual operations.
- Order these vector spaces by how many real numbers it takes to specify one element, fewest first.
- A vector space must be closed under addition and scaling and must contain a zero vector. Sort each set, taken…