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Lecture_17

# Lecture_17 - Lecture Note 17 Dr Jeff Chak-Fu WONG...

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Lecture Note 17 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1

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O RTHOGONAL C OMPLEMENTS O RTHOGONAL C OMPLEMENTS 2
Example 1 Suppose W 1 and W 2 are subsets of a vector space V . Define W 1 + W 2 . Solution W 1 + W 2 consists of all sums of w 1 a + w 2 a , where w 1 a W 1 and w 2 a W 2 : W 1 + W 2 = { w 1 a + w 2 a | w 1 a W 1 , w 2 a W 2 } . O RTHOGONAL C OMPLEMENTS 3

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Example 2 Suppose W 1 and W 2 are subspaces of a vector space V . Show that W 1 + W 2 is a subspace of V . Solution Since W 1 and W 2 are subspaces, 0 W 1 and 0 W 2 . Hence 0 = 0 + 0 W 1 + W 2 . Suppose v a , v b W 1 + W 2 . Then there exist w 1 a , w 1 b W 1 and w 2 a , w 2 b W 2 such that v a = w 1 a + w 2 a and v b = w 1 b + w 2 b . Since W 1 and W 2 are subspaces, w 1 a + w 1 b W 1 and w 2 a + w 2 b W 2 , and for any scalar c , c w 1 a W 1 and c w 2 a W 2 . Accordingly, v a + v b = ( w 1 a + w 2 a )+( w 1 b + w 2 b ) = ( w 1 a + w 1 b )+( w 2 a + w 2 b ) W 1 + W 2 and for any scalar c , c v a = c ( w 1 a + w 1 b ) = c w 1 a + c w 1 b W 1 + W 2 . Thus W 1 + W 2 is a subspace of V O RTHOGONAL C OMPLEMENTS 4
Example 3 Define the direct sum V = W 1 W 2 . Solution The vector space V is said to be the direct sum of its subspaces W 1 and W 2 , denoted by V = W 1 + W 2 if every vector v V can be written in one and only one way as v = w 1 + w 2 , where w 1 W 1 and w 2 W 2 O RTHOGONAL C OMPLEMENTS 5

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Example 4 The vector space V is the direct sum of its subspaces W 1 and W 2 , i.e., V = W 1 W 2 if and only if 1. V = W 1 + W 2 2. W 1 W 2 = { 0 } Solution ( ) Suppose V = W 1 W 2 . Then any v V can uniquely written in the form v = w 1 + w 2 , where w 1 W 1 and w 2 W 2 . Thus, in particular, V = W 1 + W 2 . Now suppose v = W 1 W 2 . Then v = v + 0 , where v W 1 , 0 W 2 and v = 0 + v , where 0 W 1 , v W 2 Since such a sum for v must be unique, v = 0 . Accordingly, W 1 W 2 = { 0 } . O RTHOGONAL C OMPLEMENTS 6
( ) On the other hand, suppose V = W 1 + W 2 and W 1 W 2 = { 0 } . Let v V . Since V = W 1 + W 2 , there also exist w 1 W 1 and w 2 W 2 such that v = w 1 + w 2 . We need to show that a sum is unique. Suppose also that v = w * 1 + w * 2 , where w * 1 W 1 and w * 2 W 2 . Then w 1 + w 2 = w * 1 + w * 2 and so w 1 - w * 1 = w 2 - w * 2 . But w 1 - w * 1 W 1 = w 2 - w * 2 W 2 ; hence by W 1 W 2 = { 0 } , w 1 - w * 1 = 0 , w 2 - w * 2 = 0 and so w 1 = w * 1 and w 2 = w * 2 . Thus such a sum for v V is unique and V = W 1 W 2 . O RTHOGONAL C OMPLEMENTS 7

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Let W 1 and W 2 be subspaces of a vector space V . Let W 1 + W 2 be the set of all vectors v in V such that v = w 1 + w 2 , where w 1 is in W 1 and w 2 is in W 2 . In Example 1 and Example 2, we show that W 1 + W 2 are a subspace of V . In Example 3 and Example 4, if V = W 1 + W 2 and W 1 W 2 = { 0 } , then V is the direct sum of W 1 and W 2 and we write V = W 1 W 2 .
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Lecture_17 - Lecture Note 17 Dr Jeff Chak-Fu WONG...

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