24.2. Examples of Vector Spaces

Try these first

1. 1.

   What is an n-tuple?

   Hint

   Think of examples like (1, 2, 3) or (x, y).

2. 2.

   Define closure property in the context of vector spaces.

   Hint

   Consider addition of two vectors.

3. 3.

   What does ℝⁿ represent?

   • A simple graph
   • n-dimensional real space
   • A type of function

   Hint

   Consider what the notation implies about dimensions.

4. 4.

   Are all real-valued functions a subset of vector spaces?

   • True
   • False

   Hint

   Evaluate if functions can be added or scaled.

5. 5.

   Show that the set of all continuous real-valued functions forms a vector space.

   Hint

   Consider the properties of continuity and addition.

6. 6.

   Determine if the set { f(x) = x³, g(x) = 2x² } is a vector space under standard polynomial operations.

   Hint

   Focus on the definition of polynomial spaces.