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lect5_5 - NOTES ON SECTION 5.5 5.6 MA265, SECTIONS 41, 52...

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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
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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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lect5_5 - NOTES ON SECTION 5.5 5.6 MA265, SECTIONS 41, 52...

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