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# lect5_5 - NOTES ON SECTION 5.5 5.6 MA265 Some key words...

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NOTES ON SECTION 5 . 5 - 5 . 6 MA265, SECTIONS 41, 52 Some key words complement orthogonal complement projection Fourier Series 1. Orthogonal complement Deﬁnition 1. Suppose W is a subspace of an inner product space ( V, ( , )) . The set W = { v V : for all w W, ( w,v ) = 0 } is called the orthogonal complement to W . I’ll illustrate this with a very simple example Example 2. Let W be the subspace of R 3 given by W = { ( x,y,z ) where x + y + z = 0 } Then W is the line through the normal vector W = span { (1 , 1 , 1) } Example 3. This time let W be the subspace of R 3 given by W = span { (1 , 1 , 1) } Then W is the plane W = { ( x,y,z ) where x + y + z = 0 } The examples suggest that ( W ) = W , which is actually true in general (but requires proof!). Deﬁnition 4. Suppose A is a subspace of a vector space V . A complement to A is another subspace B such that dim A + dim B = dim V A B = { 0 } . One might also say that A,B are complementary subspaces and so on. We denote this relationship as V = A B . Proposition 5. A,B are complements of each other if and only if for every v V can be written uniquely as v = a + b where a A,b B Proposition 6. If W is a subspace, then W is itself a (diﬀerent!) subspace of V . Moreovor, W is a complement to W , i.e. V = W W . ( W ) = W Every vector v V can be UNIQUELY written as a sum v = w + w where w W,w W . 1

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2. Relations between fundamental subspaces of A Theorem 7. If A is a m × n matrix, then null( A ) = (row( A )) null( A T ) = (col( A )) This gives us an eﬀective way to ﬁnd the orthgonal complement of subspace W of R n which is presented to us as a null space or a span: If W = span { v 1 ,...,v k } , then let A be the matrix whose columns are the vectors v i , so that W is the column space of A ; then W = null( A T ). If W = null( A ) for some matrix A , then the orthogonal complement to W is W = row( A ) (strictly speaking you then have to reshape the vectors to be column vectors so that the result is in the same vector space you started out with). See the textbook’s Example 4. 2
3. Orthogonal Projections Deﬁnition 8. If W is a subspace of an inner product space V , then we deﬁne for any vector v V the orthogonal projection of V into W by proj W ( v ) = w where w is the unique vector w W such that v = w + w for some choice of w W . In practice, to compute projections one uses an orthogonal basis for

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## This note was uploaded on 03/31/2012 for the course MA 265 taught by Professor Bens during the Spring '08 term at Purdue.

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lect5_5 - NOTES ON SECTION 5.5 5.6 MA265 Some key words...

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