This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Homework 12  Solutions 6.2 Problem 7 z ∈ W ⊥ if and only if h z, v i = 0 for all v ∈ W . Writing v as a linear combination of vectors from β , we see that this is equivalent to h z, X j α j v j i = 0 This happens if and only if h z, v j i = 0 for all j . 6.2 Problem 11 The ij th etry of AA ∗ is the inner product of the j th and i th rows od A . Therefore, AA ∗ = I if and only if h v j , v k i = δ ij whete δ ij is the Kronecker delta (See page 89). this proves that the rows form and ONB of C n . 6.2 Problem 13 (a) Let x ∈ S ⊥ . Then h x, y i = 0 for all y ∈ S , since S ⊆ S . Therefore, S ⊥ ⊆ S ⊥ . (b) Let x ∈ S . For any y ∈ S ⊥ , h x, y i = 0. Therefore, y ∈ ( S ⊥ ) ⊥ , proving S ⊆ ( S ⊥ ) ⊥ . Since S ⊥ is a subspace even if S is not, taking the span on both side of the above equation, the proof is complete. (c) By (b), W ⊆ ( W ⊥ ) ⊥ . Suppose x ∈ ( W ⊥ ) ⊥ . Then, for any z ∈ W ⊥ , h x, z i = 0. On the other hand if x / ∈...
View
Full
Document
This note was uploaded on 01/26/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.
 Spring '10
 FUCKHEAD

Click to edit the document details