Lecturenotes6-8April2011

Lecturenotes6-8April2011 - Gram-Schmidt Process This...

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Gram-Schmidt Process This process consists of steps that describes how to obtain an orthonormal basis for any ﬁnite dimensional inner products. Let V be any nonzero ﬁnite dimensional inner product space and suppose that { u 1 ,u 2 ,...,u n } is any basis for V. We will form an orthogonal basis from this basis say { v 1 ,v 2 ,...,v n } Step 1: Let v 1 = u 1 Step 2: Produce a vector that is orthogonal to v 1 . Hence v 2 = u 2 - proj v 1 u 2 = u 2 - <u 2 ,v 1 > || v 1 || 2 v 1 is orthogonal to v 1 . Step 3: Produce a vector that is orthogonal to both v 1 and v 2 , equivalently, a vector that is orthogonal to the space W 2 spanned by v 1 and v 2 . Hence v 3 = u 3 - proj W 2 u 3 = u 3 - <u 3 ,v 1 > || v 1 || 2 v 1 - <u 3 ,v 2 > || v 2 || 2 v 2 is orthogonal to both v 1 and v 2 . We will continue this way untill we produce n vectors that is v n = u n - proj W n - 1 u n = u n - < u n ,v 1 > || v 1 || 2 v 1 - < u n ,v 2 > || v 2 || 2 v 2 - ... - < u n ,v n - 1 > || v n - 1 || 2 v n - 1 Then { v 1 ,v 2 ,...,v n } will be an orthogonal basis of V and { v 1 || v 1 || , v 2 || v 2 || ,..., v n || v n || } will be an orthonormal basis of V. Example: Let R 3 have the Euclidean inner product. Use Gram-Schmidt process to transform the basis { (1 , 1 , 1) , ( - 1 , 1 , 0) , (1 , 2 , 1) } into an orthonormal basis. 1

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Solution: Let v 1 = (1 , 1 , 1) , then v 2 = ( - 1 , 1 , 0) - proj v 1 ( - 1 , 1 , 0) = ( - 1 , 1 , 0) and v 3 = (1 , 2 , 1) - proj v 1 (1 , 2 , 1) - proj v 2 (1 , 2 , 1) = ( 1 6 , 1 6 , - 1 3 ) . || v 1 || = 3 , || v 2 || = 2 , || v 3 || = q 1 6 ⇒ { ( 1 3 , 1 3 , 1 3 ) , ( - 1 2 , 1 2 , 0) , ( 1 6 , 1 6 , - 6 3 ) } is an orthonormal basis. Example: Consider the vector space M 22 of 2 × 2 real matrices with an inner product deﬁned as < A,B > = tr ( A T B ) where A and B are 2 × 2 matrices. Let S= { ± 1 0 0 0 ² , ± 0 0 1 0 ² , ± 0 0 1 1 ² } be a basis of a subspace W of inner product space M 22 . Find an orthogonal basis for W by Gram-Schmidt process.
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Lecturenotes6-8April2011 - Gram-Schmidt Process This...

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