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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.
C. When you use sources, include all in-text citations and references in APA format.
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