Newberger Math 247 Spring 03
Homework solutions: Section 2.8 #31-36
31. Suppose Fis a 5 ×5 matrix whose column space is not equal to
R5. What can you say about Nul F?
Start your explanations with the assumptions (the suppose part).
Since the column space of Fis not equal to R5,Fdoes not have a
pivot in every row. Since Fis square, we know Falso does not have
a pivot in every column. This means that the homogeneous equation
Fx=0has non-trivial solutions. Since Nul Fis the solution set to
the homogeneous equation, we can deduce that Nul Fcontains nonzero
vectors, or in other words, Nul F6={0}.
32. If Ris a 6 ×6 matrix and Nul Ris not the zero subspace, what
can you say about Col R?
Start your explanation with the assumptions (the if part). Since
Nul Ris not the zero subspace, we know that the homogeneous equation
Rx=0has nontrivial solutions. This means that Rdoes not have a
pivot in every column. Since Ris square, this also means that Rdoes
not have a pivot in every row, so the columns of Rdo not span all of
R6. Thus Col Ris not equal to R6.
33. If Qis a 4 ×4 matrix and Col Q=R4, what can you say about
solutions of equations of the form Qx=bfor bin R4?
Start with the assumptions (the if part). Since Col Q=R4, the
span of the columns of Qis all of R4, so Qhas a pivot in every row.
Furthermore, since Qis square, Qalso has a pivot in every column.
This means that Qx=bhas a unique solution for all bin R4. Keep
straight in your mind that a pivot in every row means the solution
always exists, and a pivot in every column means that when there is a
solution, that solution is unique.
34. If Pis a 5 ×5 matrix and Nul Pis the zero subspace, what can
you say about solutions of the equations of the form Px=bfor bin
R5?
Start with the assumptions (the if part). Since Nul Pis the zero sub-
space, the homogeneous equation Px=0has only the trivial solution,
which means that Phas a pivot position in every column. Since Pis
square, Palso has a pivot in every row. Thus Px=bhas a unique
solution for every bin R5. Keep straight in your mind that a pivot in
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