24.13. Direct Sum of Subspaces

Try these first

1. 1.

   What does it mean for two subspaces to have a zero intersection?

   Hint

   Think about the implications of overlap between the two spaces.

2. 2.

   Define direct sum in your own words.

   Hint

   Consider the need for uniqueness in representation.

3. 3.

   For U and W to form a direct sum, what must their intersection contain?

   • All vectors
   • Only the zero vector
   • Any vectors

   Hint

   Reflect on the definition of intersection in vector spaces.

4. 4.

   True or False: Every vector in V can be expressed in more than one way as a sum of vectors from U and W in a direct sum.

   • True
   • False

   Hint

   Consider what uniqueness implies.

5. 5.

   Show that the following subspaces U = span{(1,0)} and W = span{(0,1)} form a direct sum in ℝ².

   Hint

   Visualize this in the context of the coordinate plane.

6. 6.

   Given vector spaces V, U, and W as described in an engineering context, prove that V = U ⊕ W captures all design aspects.

   Hint

   Think about the contributions each subspace makes to the overall design.