Math 417 homework 3 solutions
(Given only for problems that are not straightforward computation; contact
the instructor if you still have questions about the others.)
Problem 12,20,24
Finding things “by inspection” is of course whichever way you can. Systematic
way: compute the reduced row echelon form. Column
j
of the original matrix is
redundant if and only if the rref does not have a leading 1 in that same column
j
. Gather the nonredundant column in a list; that is a basis for the image. For
each redundant column, read a kernel vector out of the rref, as shown in class.
Problem 28
The ﬁrst three vectors are clearly independent, because each has a 1 (nonzero)
in a row where the previous ones have 0. Looking again at rows 1 to 3, the
fourth vector can be redundant only if it is 2 times ﬁrst plus 3 times second plus
4 times third vector. In that case
k
= 2
·
2 + 3
·
3 + 4
·
4 = 29. Hence the fourth
vector is linearly independent if and only if
k
6
= 29.
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 Fall '07
 ELLING
 Math, Linear Algebra, Matrices, ax, ranknullity formula

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