Download e-book for iPad: Advanced calculus by Spencer, Donald Clayton; Nickerson, Helen Kelsall

Calculus

By Spencer, Donald Clayton; Nickerson, Helen Kelsall

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Show that each vector A of V is uniquely representable as the sum of a vector of 48 ker P and a vector of im P. ker Q = im P, 1m Q projection, and that Q r I - P is also a Show that = ker P. Show that ~· PQ = QP Let 4. dim V be finite. Show that any endomorphism T of V can be expressed as a composition SP where projection and 1f JJ S is an automorphism. An endomorphism 5. = I. Show that, 1f is an involution. Q P is a J : V -> V is called an involution is a projection, then J = I - 2P P What involution corresponds to the projection = I - P?

The system aji of coefficients is usually regarded as 35 n ·rows and a rectangular array of numbers, having k columns, called a matrix, thus a 1 , a 12 ••• a 1k) a22 ••• a2k a21 ( . ... an 1 an2 ••• ank The numbers in the i th column are the components of T (Ai) with B1, ••• , Bn. (See (1 ) • ) Definition. 1. having bases T : V -> W be linear. B1 , •.. , Bn and A1 , ••• , Ak The matrix (a ji) respectively, and of coefficients in (1) (or in (5)) defines the matrix representation of to the given choice of bases for If Any n x k of a corresponding w.

Then on the left or on the right. Proposition. (ST)- 1 = T-ls- 1 Proof. 7. 1) of a- 1T- 1 • Exercises §10. 1. = T(V). U of V which T trans- Show that each parallel (see U intersects each parallel of ker T in precisely one vector: 2. e. 3. if PP = P. An endomorphism Show that P : V -> V is called a projection ker P n im P = TIV. Show that each vector A of V is uniquely representable as the sum of a vector of 48 ker P and a vector of im P. ker Q = im P, 1m Q projection, and that Q r I - P is also a Show that = ker P.

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Advanced calculus by Spencer, Donald Clayton; Nickerson, Helen Kelsall


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