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HWSolutions2.8

# HWSolutions2.8 - Newberger Math 247 Spring 03 Homework...

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Newberger Math 247 Spring 03 Homework solutions: Section 2.8 #31-36 31. Suppose F is a 5 × 5 matrix whose column space is not equal to R 5 . What can you say about Nul F ? Start your explanations with the assumptions (the suppose part). Since the column space of F is not equal to R 5 , F does not have a pivot in every row. Since F is square, we know F also does not have a pivot in every column. This means that the homogeneous equation F x = 0 has non-trivial solutions. Since Nul F is the solution set to the homogeneous equation, we can deduce that Nul F contains nonzero vectors, or in other words, Nul F 6 = { 0 } . 32. If R is a 6 × 6 matrix and Nul R is not the zero subspace, what can you say about Col R ? Start your explanation with the assumptions (the if part). Since Nul R is not the zero subspace, we know that the homogeneous equation R x = 0 has nontrivial solutions. This means that R does not have a pivot in every column. Since R is square, this also means that R does not have a pivot in every row, so the columns of R

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HWSolutions2.8 - Newberger Math 247 Spring 03 Homework...

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