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    Vector spaces

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    The set of elements belonging to R^2 is usually denoted as {(a, b) | a, b ∈ R}. Combining elements within this set under the operations of addition and scalar multiplication should use the following notation:
     
    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.
     
    Assignment:
     
    Write an explanation of vector space where you:
     
    1) Provide a mathematical definition for a vector space.
     
    2) Indicate whether R^2 is a vector space.
    * Justify assertions by applying the provided mathematical definition for a vector space.
     
    3) Determine whether R^2 is spanned by (1, 1) and (3, 2) (show all work).
     
    4) Define a nontrivial subspace of R^2 (show all work).

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    Vector spaces are clarified in this solution.

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