24. Vector Space

Try these first

1. 1.

   Define a vector space and its basic properties.

   Hint

   Think about operations and the necessary rules that govern them.

2. 2.

   What is a subspace? Give an example.

   Hint

   Recall the properties that validate a subspace.

3. 3.

   What is a vector space?

   • A structure with addition and multiplication
   • Only a set of numbers
   • Any collection of points

   Hint

   Recall the operations that define a vector space.

4. 4.

   True or False: A subspace can include only some of the vectors in a vector space.

   • True
   • False

   Hint

   Think about the properties that define a subspace.

5. 5.

   Find and prove a basis for the vector space spanned by the vectors (3, 6) and (1, 2).

   Hint

   Consider how one vector can be formed from the other.

6. 6.

   Prove that the set W = {(x, y, z) ∈ R^3 | x + 2y + 3z = 0} is a subspace of R^3.

   Hint

   You need to satisfy all three conditions for subspaces.