21.8. Vector Spaces and Subspaces

Try these first

1. 1.

   Define a vector space.

   Hint

   Think about the operations we perform on vectors.

2. 2.

   What is a subspace?

   Hint

   What must the subset satisfy?

3. 3.

   What is a vector space?

   • A set without operations
   • A set of objects
   • A set of vectors closed under addition and scalar multiplication

   Hint

   Think about what defines a vector space.

4. 4.

   Is a subspace a vector space?

   • True
   • False

   Hint

   Recall what a subset must satisfy.

5. 5.

   Given three vectors in R^3, determine if they form a basis. Show your calculations.

   Hint

   Use row reduction to check for independence.

6. 6.

   You have a vector space of dimension 5. If you remove one vector that is part of the basis, what happens to the dimension? Explain your reasoning.

   Hint

   How do basis vectors relate to the definition of dimension?