Solutions to Assignment 8
Math 217, Fall 2002
4.6.26
If
A
is 6
×
4, what is the smallest possible dimension of Nul(
A
)?
Well, we know that rank(
A
)+dim(Nul(
A
)) = 4. It is possible that rank(
A
) = 4,
that is, that dim(Col(
A
)) = 4 because
A
is 6
×
4.
We conclude the smallest
possible dimension of Nul(
A
) is zero.
4.6.20
Suppose a nonhomogeneous system of nine linear equations in ten unknowns
has a solution for all possible constants on the right sides of the equations. Is
it possible to find two nonzero solutions of the associated homogeneous system
that are not multiples of each other? Discuss.
Again, we know that rank(
A
) + dim(Nul(
A
)) = 10. If the system is consistent
for all possible constants on the right side of the equation, then the matrix of
coefficients must have a pivot in every row. As the matrix is 9
×
10, this means
that the matrix must have 9 pivots. Thus there are 9 pivot columns, and so the
rank(
A
) = 9. It must be the case that dim(Nul(
A
)) = 1, and we conclude that
a basis for the vector space of solutions to the homogeneous system of equations
contains only one element. This means that every nonzero solution is a multiple
of the one basis element. So it is not possible to find two nonzero vectors which
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 Spring '10
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 Math, Linear Algebra

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