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# hw05 - A of a square matrix A is deﬁned to be the sum of...

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Math 110—Linear Algebra Fall 2009, Haiman Problem Set 5 Due Monday, Oct. 5 at the beginning of lecture. 1. Given vector spaces V and W over a field F , and subspaces V 0 V , W 0 W , show that each of the following sets of linear transformations is a subspace of L ( V, W ): (a) { T : V W such that V 0 N ( T ) } (b) { T : V W such that R ( T ) W 0 } 2. Let V be a finite-dimensional vector space and T : V V a linear transformation. Prove that if T 2 = 0, then dim( R ( T )) dim( V ) / 2 dim( N ( T )). 3. Section 2.3, Exercise 9. 4. Recall that the trace tr( A
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Unformatted text preview: A ) of a square matrix A is deﬁned to be the sum of the diagonal entries A ii . Show that for all A ∈ M m × n ( F ) and B ∈ M n × m ( F ), we have tr( AB ) = tr( BA ) . (Note that AB is m × m and BA is n × n , so they are square matrices.) 5. Let A be a matrix over F . Prove that rank( L A ) = 1 if and only if there exist a non-zero row vector X and a non-zero column vector Y such that A = Y X ....
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