23.1. Vector Spaces and Basis (Recap)

Try these first

1. 1.

   Define a vector space.

   Hint

   Think about the operations you can perform within it.

2. 2.

   What is a basis of a vector space?

   Hint

   Consider the relationship between spanning and independence.

3. 3.

   What is a vector space?

   • A collection of vectors unable to add
   • A closed set under addition and scalar multiplication
   • Any collection of numbers

   Hint

   Remember the definition.

4. 4.

   Is every basis a linearly independent set?

   • True
   • False

   Hint

   Consider the definition of basis.

5. 5.

   Consider the set of vectors A = {(1,2), (2,4), (3,6)}. Prove if A is linearly independent or dependent.

   Hint

   Use the properties of scaling factors.

6. 6.

   In a space R^4, you need to determine if the set { (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,1,1,1) } is independent.

   Hint

   Consider the dimension relation.