2. Let W= = {(x³-5, y-1,z²+4) € R³|x, y, z E R}. Is W a vector subspace of R³? Justify your reasoning.

A vector space is a fundamental concept in linear algebra, and it provides a mathematical structure that allows for the study of vectors and vector operations. A vector space is defined as a set of vectors equipped with two operations: vector addition and scalar multiplication, which satisfy certain properties. Here's the formal definition:

Let V be a non-empty set of objects (called vectors) over a field F (usually the real numbers, denoted as ℝ, or the complex numbers, denoted as ℂ). A vector space is a pair (V, F) along with two operations, vector addition (+) and scalar multiplication (·), such that the following axioms are satisfied for all vectors u, v, and w in V, and all scalars (elements of F), a and b:

1. Closure under vector addition: If u and v are in V, then their sum u + v is also in V.
2. Commutativity of vector addition: u + v = v + u for all u and v in V.
3. Associativity of vector addition: (u + v) + w = u + (v + w) for all u, v, and w in V.
4. Existence of a zero vector: There exists a vector, denoted as 0, such that for any vector u in V, u + 0 = u.
5. Existence of additive inverses: For every vector u in V, there exists a vector v in V such that u + v = 0. This v is denoted as -u.
6. Closure under scalar multiplication: If u is in V and a is in F, then the product a·u is also in V.
7. Compatibility of scalar multiplication with field multiplication: a·(b·u) = (a·b)·u for all a, b in F and u in V.
8. Distributivity of scalar multiplication over vector addition: a·(u + v) = a·u + a·v for all a in F and u, v in V.
9. Distributivity of scalar multiplication over field addition: (a + b)·u = a·u + b·u for all a, b in F and u in V.
10. Identity element for scalar multiplication: There exists a scalar 1 in F such that 1·u = u for all u in V.

These axioms ensure that a vector space is a well-behaved algebraic structure that generalizes the notion of vectors and their manipulation in a consistent manner. Vector spaces are the foundation for many mathematical and scientific disciplines, including linear algebra, physics, and engineering. They provide a framework for studying linear transformations, systems of linear equations, and many other important mathematical concepts.