MAT1341-L14-OrthogonalSpaces0

# MAT1341-L14-OrthogonalSpaces0 - ORTHOGONAL SPACES JOS ´ E...

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Unformatted text preview: ORTHOGONAL SPACES JOS ´ E MALAG ´ ON-L ´ OPEZ Now we will get different descriptions of R n . The first case is in terms of orthogonal subspaces. Orthogonal Complement Definition 1. Let V be a subspace of R n . The orthogonal comple- ment of V , denoted as V ⊥ , is the set of all the vectors in R n that are orthogonal to all the vectors in V . In other words, for any vectorv in V , and any vectorw in V ⊥ , we have vectorv • vectorw = 0. Remark 2 . Notice that if { vectorv 1 , . . . ,vectorv r } is a basis for V , then vectorw is in V ⊥ if and only if vectorw • vectorv i = 0, for 1 ≤ i ≤ r . Theorem 3. V ⊥ is a subspace of R n . Proof . Let { vectorv 1 , . . . ,vectorv r } be a basis for V . Let vectorw 1 , vectorw 2 be any two vectors in V ⊥ , and let α and β be any two scalars. Then vectorv i • ( αvectorw 1 + β vectorw 2 ) = α ( vectorv i • vectorw 1 ) + β ( vectorv i • vectorw 2 ) = α 0 + β 0 = 0 , for any 1 ≤ i ≤ r . Q.E.D. Example 4. If A is any m × n matrix, and V = Row( A ), we have by definition that V ⊥ = (Row( A )) ⊥ = Null( A ). Remark 5 . From the definition of the transpose of a matrix we know that Col( A T ) = Row( A ). Thus, we have that Col ( A T ) ⊥ = Row( A ) ⊥ = (Null( A )) , which is equivalent to having Col ( A ) ⊥ = Row ( A T ) ⊥...
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MAT1341-L14-OrthogonalSpaces0 - ORTHOGONAL SPACES JOS ´ E...

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