Respuesta :
The set of all functions x (t) = a1 sin wt + a2 coswt is a vector space.
What is vector space?
Vector space is a mathematical concept that describes a set of vectors in which operations such as vector addition, scalar multiplication, and vector scalar multiplication can be performed. Vector spaces are usually equipped with an inner product, which allows for the definition of the length of a vector and the angle between two vectors. Vector spaces are commonly used to model problems in linear algebra, geometry, and engineering. Examples of vector spaces include Euclidean and affine spaces, as well as Hilbert and Banach spaces. Vector spaces are typically associated with a field, such as the real numbers or complex numbers, which allows for the definition of a vector space over the field.
A vector space is a set of elements that is closed under addition and scalar multiplication. This means that if two elements in the set, x1 (t) and x2 (t), are added together and the result is also in the set, and if any element in the set is multiplied by a scalar and the result is also in the set.
In this mass-spring system, we have a set of functions x (t) = a1 sin wt + a2 coswt.
We can see that the set is closed under addition because if we add two functions x1 (t) = a1 sin wt + a2 coswt and x2 (t) = b1 sin wt + b2 coswt, we get x3 (t) = (a1+b1) sin wt + (a2+b2) coswt which is also an element of the set. We can also see that the set is closed under scalar multiplication because if we multiply any function x (t) = a1 sin wt + a2 coswt by a scalar c, we get c x (t) = ca1 sin wt + ca2 coswt which is also an element of the set.
Therefore, the set of all functions x (t) = a1 sin wt + a2 coswt is a vector space.
To learn more about vector space
https://brainly.com/question/29757201
#SPJ4
