Exercise 8. Let W be a subspace of C of dimension k. Show that W possesses...

 

Transcribed Image Text:

Exercise 8. Let W be a subspace of C" of dimension k. Show that W possesses a basis {b1,..., bx} which is orthonormal with respect to the Hermitian inner product (i.e. b; b; = 0 for i + j, and bib; = 1 for each i). Hint: Start with any basis {c1,...,Ck} of W, and adapt it to a basis of mutually orthogonal vectors using some version of the Gram-Schmidt orthogonalizaton process. Write bị = cị and show that some linear combination b2 of bị and c2 is orthogonal to b1. Then show that some linear combination b2 of bi, b2 and c3 is orthogonal to both bị and b2. Build up a basis this way - once you have a basis whose elements are all orthogonal to each other, you can scale the elements to ensure bị bị = 1 for each i. Exercise 8. Let W be a subspace of C" of dimension k. Show that W possesses a basis {b1,..., bx} which is orthonormal with respect to the Hermitian inner product (i.e. b; b; = 0 for i + j, and bib; = 1 for each i). Hint: Start with any basis {c1,...,Ck} of W, and adapt it to a basis of mutually orthogonal vectors using some version of the Gram-Schmidt orthogonalizaton process. Write bị = cị and show that some linear combination b2 of bị and c2 is orthogonal to b1. Then show that some linear combination b2 of bi, b2 and c3 is orthogonal to both bị and b2. Build up a basis this way - once you have a basis whose elements are all orthogonal to each other, you can scale the elements to ensure bị bị = 1 for each i.

Answer rating: 100% (QA)

Orthogonal vecto... View the full answer