Reference no: EM13801508

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.

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