26. Vector Spaces

Try these first

1. 1.

   What is a vector space?

   Hint

   Think about the basic operations involved.

2. 2.

   List the two operations that define a vector space.

   Hint

   Recall the foundational operations of vector spaces.

3. 3.

   What is a vector space defined by?

   • Only addition
   • Only scalar multiplication
   • Addition and scalar multiplication

   Hint

   Think about the basic definition of vector spaces.

4. 4.

   True or False: All subsets of a vector space are also vector spaces.

   • True
   • False

   Hint

   Consider the definitions of subspaces.

5. 5.

   Given three vectors in ℝ³, determine if they are linearly independent or dependent. Provide an explanation.

   Hint

   Look for coefficients that are not all zero.

6. 6.

   Define the null space of a given matrix and explain its significance.

   Hint

   Think about how null spaces relate to linear systems.