26.3. Subspaces

Try these first

1. 1.

   What are the three conditions for a subset to be a subspace?

   Hint

   Think about the defining properties of a vector space.

2. 2.

   Is the set of vectors { (1,0), (0,1), (1,1) } a subspace of ℝ²?

   Hint

   Verify if zero vector is included and if adding two members gives another member.

3. 3.

   What is a requirement for a subset W to be a subspace of V?

   • It must contain at least two vectors
   • It must be closed under vector addition
   • It must not include the zero vector

   Hint

   Think of the operations allowed within the vector space.

4. 4.

   True or False: All lines in ℝ² are subspaces.

   • True
   • False

   Hint

   Recall the specific properties of subspaces you've learned.

5. 5.

   Prove that the span of any set of vectors is a subspace.

   Hint

   Start by considering what happens when you combine vectors in the span.

6. 6.

   Find two vectors in ℝ² that form a line through the origin and demonstrate the closure properties.

   Hint

   Visualize this on a graph for clarity.