section10 - 11/04/11 EE221A Section 10 1 Direct sum of...

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EE221A Section 10 11/04/11 1 Direct sum of subspaces Exercise 1. Show that if V = V 1 V 2 ⊕ ··· ⊕ V n , then V i V j = { θ } for i 6 = j . Exercise 2. Let M and N be two subspaces of V . Let { m 1 ,...,m p } be a basis for M , and { n 1 ,...,n k } be a basis for N . Show that V = M N if and only if { m 1 ,...,m p ,n 1 ,...,n k } is a basis of V . Exercise 3. Show that if V = M N , we can construct a linear operator P (called the projection on M parallel to N ) satisfying P 2 = P (idempotent), R ( P ) = M , and N ( P ) = N . 2 Jordan form Recall: Minimal polynomial ˆ ψ A ( s ) of A is the polynomial of least degree such that ˆ ψ A ( A ) = θ n × n . The minimal polynomial divides the characteristic polynomial : This means that ˆ ψ A ( s ) = ( s - λ 1 ) m 1 ( s - λ 2 ) m 2 ··· ( s - λ σ ) m σ , where σ is the number of distinct eigenvalues of A , and m i d i , i = 1 ,...,σ , where d i is the degree of the i -th distinct eigenvalue in the characteristic polynomial. Theorem:
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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section10 - 11/04/11 EE221A Section 10 1 Direct sum of...

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