Collection of specific characteristics and properties

Question

Questions:

Vector spaces possess a collection of specific characteristics and properties.

The set of elements belonging to R² 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 R² is a vector space.

* Justify assertions by applying the provided mathematical definition for a vector space.

3)    Determine whether R² is spanned by (1, 1) and (3, 2) (show all work).

4)    Define a nontrivial subspace of R² (show all work).