Vector spaces

Introduction:Vector spaces possess a collection of specific characteristics and properties. Use the definitions in the attached “Definitions” to complete this task.Define the elements belonging to R2 as {(a, b) | a, b ∈ R}. Combining elements within this set under the operations of vector addition and scalar multiplication should use the following notation:Vector Addition Example: (–2, 10) + (–5, 0) = (–2 – 5, 10 + 0) = (–7, 10)Scalar Multiplication Example: –10 × (1, –7) = (–10 × 1, –10 × –7) = (–10, 70), where –10 is a scalar.Under these definitions for the operations, it can be rigorously proven that R2 is a vector space.Given:Laws 1 and 2 in the attached “Definitions” are true.Requirements:Provide a written explanation (suggested length of 2–4 pages) of why R2 is a vector space in which you do the following:A. Prove the truth of Laws 3 through 10 of the provided mathematical definition for a vector space.B. Give an example of a subset of R2 that is a nontrivial subspace of R2, showing all work.Task 2 DefinitionsThe Definition of a Vector Space:
A vector space is a set V of elements (called vectors) together with two operations, vector addition and scalar multiplication, satisfying the following 10 laws (for all vectors X, Y, and Z in V and all (real) scalars r and s).Laws for Addition
1. Closure under addition
If X and Y are any two vectors in V, then X + Y ∈ V.
2. Associative law
(X + Y) + Z = X + (Y + Z)
3. Commutative law
X + Y = Y + X
4. Additive identity law
There is a vector in V, denote 0 such that X + 0 = X where 0 is called the zero vector.
5. Additive inverse law
For every X ∈ V there is a vector –X such that X + (–X) = 0, where –X is called the additive inverse of X.Laws for Scalar Multiplication
6. Closure under scalar multiplication
If X is any vector in V and r is any real scalar, then rX ∈ V.
7. Associative law
(rs)X = r(sX)
8. Distributive Property of Scalar Multiplication Over Vector Addition
r(X + Y) = rX + rY
9. Distributive Property of Scalar Multiplication Over Scalar Addition
(r + s)X = rX + sX
10. Unit law
1X = XThe Definition of a Nontrivial Subspace of a Vector Space:
Any subspace W of V that is neither V nor {0} is a nontrivial subspace (note that the set W needs to be proven a subspace before it can be proven to be a nontrivial subspace