27.9. Orthogonal Complement

Try these first

1. 1.

   What is the definition of the orthogonal complement?

   Hint

   Think about the vectors that satisfy the zero inner product condition.

2. 2.

   If W is a subspace of a vector space V, what does W⊥ represent?

   Hint

   Consider what 'orthogonal' means in terms of the inner product.

3. 3.

   What does the orthogonal complement W⊥ consist of?

   • All vectors in V
   • Vectors orthogonal to W
   • The zero vector only

   Hint

   Consider how orthogonality relates to inner products.

4. 4.

   True or False: The orthogonal complement of a subspace W can never include the zero vector.

   • True
   • False

   Hint

   Think about the property of the zero vector with respect to inner products.

5. 5.

   Consider a vector space V = R² and a subspace W = span{(1, 1)}. Find the orthogonal complement W⊥ and describe its geometric representation.

   Hint

   Use the definition of orthogonality and visualize it graphically.

6. 6.

   Let W be the set of vectors {(1, 2, 0), (1, 0, 1)} in R³. Determine W⊥ and provide a justification for your findings.

   Hint

   You may want to set up a system of linear equations based on the orthogonality conditions.