24.7. Dimension

Try these first

1. 1.

   What is the dimension of ℝ³?

   Hint

   Think about how many directions you can move in 3D space.

2. 2.

   Is a vector space with a basis containing 5 vectors finite-dimensional?

   Hint

   Consider the definition of finite-dimensional spaces.

3. 3.

   What is the dimension of the vector space ℝ⁵?

   • 3
   • 4
   • 5
   • 2

   Hint

   Recall how many independent vectors are required to describe this space.

4. 4.

   True or False: A vector space can have only one finite basis.

   • True
   • False

   Hint

   Think about the nature of linear independence.

5. 5.

   Show that the set of functions {1, x, x^2, ..., x^n} is a basis for the space of polynomials of degree ≤ n.

   Hint

   Explore how polynomial degrees restrict combinations.

6. 6.

   Discuss why the dimension of the vector space of all sequences of real numbers is infinite.

   Hint

   Think about the nature of sequences and how they can continue indefinitely.