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Unformatted text preview: Extend the given vector to an arbitrary basis of R 3 and perform GramSchmidt, starting with the given vector as the ±rst basis vector. 6.4 Problem 17 Since A is unitary, its columns { v 1 , . . . , v n } form an ONB of C n . Consider the product A ∗ A = I . We can compute the (1 , j ) entry of the product by expressing it as the inner product of v 1 and v j . Using the fact that A is upper triangular, we now deduce that all entries in the ±rst row of A are zero ecept for a 11 . A similar argument using the inner product of v j and v k for k > j is used to prove every entry in the j th row of A are zero except for the entry on the diagonal. (Hint: Use induction on j and k )....
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 Spring '10
 FUCKHEAD
 Linear Algebra, finitedimensional vector space, invertible linear transformation, ﬁrst basis vector

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