26.1. Definition of a Vector Space

Try these first

1. 1.

   What is a vector space?

   Hint

   Look for the definition in your notes.

2. 2.

   What is the additive identity in a vector space?

   Hint

   Think about what element doesn't change other elements when added.

3. 3.

   What operation must a vector space support?

   • Vector subtraction
   • Vector addition
   • Multiplication by vectors

   Hint

   Think of the basic operations involved in combining vectors.

4. 4.

   True or False: The scalar multiplication of a vector always remains in the same vector space.

   • True
   • False

   Hint

   Recall the properties governing scalar multiplication.

5. 5.

   Consider a set of vectors in ℝ². Prove that the sum of any two vectors in this set results in another vector in the same set (closure under addition).

   Hint

   Use the definition of vector addition.

6. 6.

   Describe how the axioms of a vector space support linear operations in a real-world application.

   Hint

   Relate back to the scenarios we discussed regarding applications in engineering.