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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 âˆˆ R}. Combining elements within this set under the operations of addition and scalar multiplication should use the following notation:
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Addition Example: (-2,Â 10)Â +Â (-5,Â 0)Â =Â (-2Â -Â 5,Â 10Â +Â 0)Â =Â (-7,Â 10)
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Scalar Multiplication Example: -10 Ã— (1, -7) = (-10 Ã— 1, -10 Ã— -7) = (-10, 70), where -10 is a scalar.
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Assignment:
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Write an explanation of vector space where you:
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1) Provide a mathematical definition for a vector space.
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2) Indicate whether R^2 is a vector space.
* Justify assertions by applying the provided mathematical definition for a vector space.
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3) Determine whether R^2 is spanned by (1,Â 1) and (3,Â 2) (show all work).
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4) Define a nontrivial subspace of R^2 (show all work).