21.8.1. Vector Space

Try these first

1. 1.

   Define vector space.

   Hint

   Remember the properties like closure and identity.

2. 2.

   What is a subspace?

   Hint

   Think of it as a 'mini vector space.'

3. 3.

   What is a defining property of a vector space?

   • Linear independence
   • Closure
   • Scalar multiplication only

   Hint

   Think about how operations are handled.

4. 4.

   True or False: A subspace must contain the zero vector.

   • True
   • False

   Hint

   Consider the origins of vector spaces.

5. 5.

   Prove that the set of all vectors of the form (a, 0) in R^2 is a subspace. Include a demonstration that it obeys all vector space rules.

   Hint

   Focus on adding two vectors and multiplying by different scalars.

6. 6.

   Given four vectors in R^3, determine if they can form a basis based on their linear independence and span. Provide steps for your reasoning.

   Hint

   Determinants can also help in checking linear independence.