Unformatted text preview: , 1]. Deﬁne an inner product on X by ( f,g ) = Z 1 f ( t ) g ( t ) dt. Let Y be the subspace of X spanned by f ,f 1 ,f 2 ,f 3 , where f k ( x ) = x k . Find an orthonormal basis for Y . (6) Let Y be a subspace of a Euclidean space X , and P Y : X → X the orthogonal projection onto Y . Prove that P * Y = P Y . (7) Show that a matrix M is orthogonal iﬀ its column vectors form an orthonormal set. (8) Let X be an ndimensional real Euclidean space, and A : X → X a linear map. Deﬁne the map f : X → X by f ( x,y ) = x T Ay . Give (with proof) necessary and suﬀcient conditions on A for f to be an inner product on X ....
View
Full Document
 Spring '08
 Staff
 Linear Algebra, Algebra, dt, 0 g, real Euclidean space, 2 0 1 g

Click to edit the document details