Printable vector space verification exercises linear algebra

📚 Topic Summary

A vector space is a set of objects (vectors) that can be added together and multiplied by scalars. To verify if a set $V$ is a vector space, you must check if it satisfies eight axioms: closure under addition and scalar multiplication, commutativity and associativity of addition, existence of an additive identity and inverse, and distributivity and associativity of scalar multiplication. These axioms ensure that the operations within the space behave predictably and consistently. For example, the set of all $m \times n$ matrices with real entries forms a vector space.

🗂️ Part A: Vocabulary

Match the term to its definition:

1. Term: Vector Space
2. Term: Scalar Multiplication
3. Term: Additive Identity
4. Term: Additive Inverse
5. Term: Closure Under Addition

1. Definition: An element that, when added to any vector, results in that same vector.
2. Definition: The property that the sum of any two vectors in the space remains within the space.
3. Definition: A set of objects satisfying eight specific axioms.
4. Definition: For every vector, there exists another vector that, when added, yields the additive identity.
5. Definition: Multiplying a vector by a scalar, resulting in another vector in the same space.

(Match the numbers from the 'Term' list with the numbers from the 'Definition' list.)

📝 Part B: Fill in the Blanks

To prove a set is a vector space, one must verify all ______ axioms. The existence of an ______ is crucial. Also, the result of ______ two vectors in the space must remain in the space to satisfy the ______ axiom. When multiplying a vector with a ______, the result must also be a vector in the space.

🤔 Part C: Critical Thinking

Explain in your own words why verifying the closure axioms is important when determining if a set is a vector space. Provide an example of a set that fails one of these axioms and explain why it's not a vector space.