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la-q4-sol

# la-q4-sol - MAS 4105 QUIZ 4 SOLUTION SPRING 2011 1[1 3 2 =...

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MAS 4105 — QUIZ 4 SOLUTION — SPRING 2011 1 . [1 + 3 + 2 = 6 pts] (i) Let A M m × n ( F ). The rank of the matrix A is defined to be the rank of the transfor- mation L A , where L A : F n -→ F m and L A ( ⃗x ) = A⃗x . (ii) Let A be an m × n matrix and B be an n × p matrix. Then L B : F p -→ F n by L B ( ⃗x ) = B⃗x , and L AB : F p -→ F m by L AB ( ⃗x ) = AB⃗x . Now suppose ⃗x N ( L B ). Then B⃗x = L B ( ⃗x ) = 0. So L AB ( ⃗x ) = AB⃗x = A ( B⃗x ) = A 0 = 0 and ⃗x N ( L AB ). Hence N ( L B ) N ( L AB ), N ( L B ) is a subspace N ( L AB ) and nullity( L B ) nullity( L AB ) . By The Dimension Theorem we have nullity( L B ) + rank( L B ) = dim( F p ) = p, nullity( L AB ) + rank( L AB ) = dim( F p ) = p, so that nullity( L B ) + rank( B ) = nullity( L AB ) + rank( AB ) , rank( B ) - rank( AB ) = nullity( L AB ) - nullity( L B ) 0 , and rank( AB ) rank( B ) . (iii) Let P = 1 0 - 1 1 1 - 2 0 0 1 M 3 × 3 ( R ) . Then 1 0 - 1 . . . 1 0 0 1 1 - 2 . . . 0 1 0 0 0 1 . . . 0 0 1 -→ 1 0 - 1 . . . 1 0 0 0 1 - 1 . . . - 1 1 0 0 0 1 . . . 0 0 1 -→ 1 0 0 .

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la-q4-sol - MAS 4105 QUIZ 4 SOLUTION SPRING 2011 1[1 3 2 =...

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