24.3. Subspace

Try these first

1. 1.

   Give an example of a zero vector in R².

   Hint

   What are the components that make a vector zero?

2. 2.

   Determine if the following set is a subspace of R²: W = {(x, y) ∈ R² : x - y = 0}.

   Hint

   Check if the zero vector (0,0) satisfies the equation.

3. 3.

   What is a key requirement for a subset to be a subspace?

   • It must contain more than one vector
   • It must contain the zero vector
   • It must be infinite-dimensional

   Hint

   Think about the foundational element shared by all vector spaces.

4. 4.

   True or False: Any linear combination of vectors in a subspace will result in a vector that is also in the subspace.

   • True
   • False

   Hint

   Consider the definition of span.

5. 5.

   Given vectors (2, 3) and (4, 6) in R², determine if their span forms a subspace of R².

   Hint

   Check if the vectors are linearly independent or dependent.

6. 6.

   Demonstrate if the intersection of two lines in R² is a subspace.

   Hint

   Consider different conditions under which the lines may or may not intersect.