21.8.2. Subspace

Try these first

1. 1.

   Define a subspace in your own words.

   Hint

   Think about closure under operations.

2. 2.

   What is meant by the basis of a subspace?

   Hint

   Consider how you can represent all vectors in the subspace.

3. 3.

   What is a subspace?

   • A larger space than a vector space
   • A subset of a vector space that is itself a vector space
   • A collection of all the linear transformations

   Hint

   Think about properties shared with the original space.

4. 4.

   True or False: Every subset of a vector space is a subspace.

   • True
   • False

   Hint

   Consider the criteria for being a subspace.

5. 5.

   Given the vectors (1, 2, 3) and (2, 4, 6), determine if they form a basis for R³ and justify why or why not.

   Hint

   Consider the independence of these vectors.

6. 6.

   Prove that the set of all vectors in R² that can be expressed as (x, 0) for any real number x forms a subspace.

   Hint

   Check that adding any two such vectors still results in a vector of the same form.