24.16. Worked Examples

Try these first

1. 1.

   Is the set W = {(x, y, z) ∈ R3: 2x + y - z = 0} a subspace? Explain.

   Hint

   Check the zero vector and the closure properties.

2. 2.

   What is a basis for W = { (x,y,z): x=0 }?

   Hint

   You need to express other variables in terms of the free ones.

3. 3.

   What is a necessary condition for a subset W to be a subspace of R3?

   • It must include the zero vector.
   • It must be closed under addition.
   • It must meet both conditions.

   Hint

   Think about the properties necessary for vector spaces.

4. 4.

   True or False: Every set of linearly independent vectors forms a basis.

   • True
   • False

   Hint

   Remember the definition of a basis.

5. 5.

   Determine if the set W = {(x, y, z) ∈ R3: 3x - y + 4z = 5} can be a subspace.

   Hint

   Check the zero vector first.

6. 6.

   Find a basis for the span of vectors {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and determine its dimension.

   Hint

   Use row reduction to assess linear independence.