Science:Math Exam Resources/Courses/MATH307/April 2006/Question 06 (a)

MATH307 April 2006

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[hide] Question 06 (a)
Determine if each of the following statements is true of false. Show reason or proof if true and show reason or counter example if false. If AB is defined, then R(AB) ⊂ R(A) and rank(AB) ≤ rank(A) (i.e. the column space of AB is contained in the column space of A and the rank of AB is at most equal to that of A.)

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[show] Solution

True Given A as an $n\times m$ and B as an $m\times q$ matrix (to be defined) then AB is an $n\times q$ matrix. A can have at the most n pivot columns, i.e. rank(A) = n and AB can have at the most n pivot columns, hence rank(AB) ≤ rank(A). Likewise, since R(AB) is also contained in the the subspace R(A) , the easiest way to see this is if given vector y if ${\vec {y}}=(AB){\vec {x}}$ then ${\vec {y}}=A(B{\vec {x}})$ hence the rank of AB is contained in the rank of A