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Unformatted text preview: Math 215 HW #6 Solutions 1. Problem 3.1.14. Show that x y is orthogonal to x + y if and only if k x k = k y k . Proof. First, suppose x y is orthogonal to x + y . Then 0 = h x y , x + y i = ( x y ) T ( x + y ) = x T x + x T y y T x y T x = h x , x i + h x , y i  h y , x ii y , y i = h x , x i  h y , y i since h x , y i = h y , x i . In other words, 0 = k x k 2 k y k 2 , so k x k 2 = k y k 2 . Since the norm of a vector can never be negative, this implies that k x k = k y k . Thus, we see that if x y is orthogonal to x + y , then k x k = k y k . On the other hand, suppose k x k = k y k . Then h x y , x + y i = ( x y ) T ( x + y ) = x T x + x T y y T x y T x = h x , x i + h x , y i  h y , x ii y , y i = h x , x i  h y , y i = k x k 2 k y k 2 = 0 , so we see that x y is orthogonal to x + y . We’ve seen that the implication goes both ways, so we conclude that x y is orthogonal to x + y if and only if k x k = k y k , as desired. 2. Problem 3.1.20. Let S be a subspace of R n . Explain what ( S ⊥ ) ⊥ = S means and why it is true. Answer: First, ( S ⊥ ) ⊥ is the orthogonal complement of S ⊥ , which is itself the orthogonal complement of S , so ( S ⊥ ) ⊥ = S means that S is the orthogonal complement of its orthogonal complement. To show that it is true, we want to show that S is contained in ( S ⊥ ) ⊥ and, conversely, that ( S ⊥ ) ⊥ is contained in S ; if we can show both containments, then the only possible conclusion is that ( S ⊥ ) ⊥ = S . To show the first containment, suppose v ∈ S and w ∈ S ⊥ . Then h v , w i = 0 1 by the definition of S ⊥ . Thus, S is certainly contained in ( S ⊥ ) ⊥ (which consists of all vectors in R n which are orthogonal to S ⊥ ). To show the other containment, suppose v ∈ ( S ⊥ ) ⊥ (meaning that v is orthogonal to all vectors in S ⊥ ); then we want to show that v ∈ S . I’m sure there must be a better way to see this, but here’s one that works. Let { u 1 , . . . , u p } be a basis for S and let { w 1 , . . . , w q } be a basis for S ⊥ . If v / ∈ S , then { u 1 , . . . , u p , v } is a linearly independent set. Since each vector in that set is orthogonal to all of S ⊥ , the set { u 1 , . . . , u p , v , w 1 , . . . , w q } is linearly independent. Since there are p + q +1 vectors in this set, this means that p + q +1 ≤ n or, equivalently, p + q ≤ n 1. On the other hand, if A is the matrix whose i th row is u T i , then the row space of A is S and the nullspace of A is S ⊥ . Since S is pdimensional, the rank of A is p , meaning that the dimension of nul( A ) = S ⊥ is q = n p . Therefore, p + q = p + ( n p ) = n, contradicting the fact that p + q ≤ n 1. From this contradiction, then, we see that, if1....
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 Spring '09
 Math, Linear Algebra, S⊥

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