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

MATH307 April 2006

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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.)

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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