Unformatted text preview: to show that rank( A T A ) = rank( A ) and null( A T A ) = null( A ) . Problem 4. Consider the linear transformation T : R 3 → R 3 deﬁned by T ( ~x ) = M ~x in terms of the 3 × 3 matrix M = 2 9 13 3 3 2 6 20 11 . a. Compute the QRfactorization of M . b. Find an orthonormal basis for the image of T . Problem 5. Let V and W be linear spaces with inner products h·i V and h·i W , respectively. A linear transformation T : V → W is said to be an isometry if it preserves lengths i.e.,  T ( f )  W =  f  V for all f ∈ V . a. Show that every isometry is a monomorphism i.e., if T is an isometry then ker( T ) = { } . b. Say V = W = R n , and that h·i V = h·i W is the dot product. Show that T is an isometry if and only if it is orthogonal....
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 Spring '08
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 Linear Algebra, Algebra, web site, course web site

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