27. Inner Product Spaces

Try these first

1. 1.

   What is an inner product space?

   Hint

   Think about how it measures angles and lengths.

2. 2.

   Define orthogonality in the context of vector spaces.

   Hint

   Relate this to geometry, where what does it mean to be at a right angle?

3. 3.

   What is the property of the Cauchy-Schwarz inequality?

   • |⟨u,v⟩| ≤ ||u|| + ||v||
   • |⟨u,v⟩| ≤ ||u||·||v||
   • ⟨u,v⟩ = ||u|| ||v||

   Hint

   Think about relationships in geometry.

4. 4.

   True or False: In an inner product space, if |⟨u,v⟩| is greater than ||u||·||v||, then u and v are linearly independent.

   • True
   • False

   Hint

   Recall the definition of linear dependence.

5. 5.

   Prove that if two vectors are orthogonal in an inner product space, their linear combination with appropriate coefficients will also remain in the span of those vectors.

   Hint

   Use the properties of linear combinations.

6. 6.

   Using the Gram–Schmidt process, convert the vectors u=(1,1,1) and v=(1,2,3) into orthonormal basis vectors.

   Hint

   Apply the iterative projection formulas carefully.