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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Solution Summary
Vector spaces are clarified in this solution.
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