Vector Spaces in Linear Algebra | Explanation and Example-2024

Vector Spaces in Linear Algebra
Mathematically, the input and output signals to an engineering system are functions. It is important in applications that these functions can be added, and multiplied by scalars. These two operations on functions have algebraic properties that are completely analogous to the operations of adding vectors in Rn and multiplying a vector by a scalar. For this reason, the set of all possible inputs (functions) is called a vector space. The mathematical foundation for systems engineering rests on vector spaces of functions.
A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars(real numbers), subject to the ten axioms (or rules) listed below. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d.
1. The sum of u and v, denoted by u +v, is in V .
2. u + v= D v + u.
3.(u +v)+w = u +(v + w).
4. There is a zero vector 0 in V such that u + 0 = u.
5. For each u in V , there is a vector -u in V such that u + (-u) = 0.
6. The scalar multiple of u by c, denoted by cu, is in V .
7. c.uv= cu cv.
8. .c (u + v)=cu + cv.
9. c.(du) =(cd).u.
10. 1u= u

A subspace of a vector space V is a subset H of V that has three properties:
a. The zero vector of V is in H.
b. H is closed under vector addition. That is, for each u and v in H, the sum u +v is in H.
c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H.

1. Let V be the first quadrant in the xy-plane; that is, let
V =[x y]
x is greater than or equal to 0, y is greater than or equal to 0
a. If u and v are in V , is u + v in V? why?
Why?
b. Find a specific vector u in V and a specific scalar c such that cu is not in V . (This is enough to show that V is not a vector space.
Let W be the union of the first and third quadrants in the xy-plane. That is, let W=[x y]
x.y=0 product of x and y is greater than or equal to 0.
Find specific vectors u and v in W such that u + v is not in W . This is enough to show that W is not a vector space.

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