You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Mathematical Expression Editor
[?]
Challenge Problems for Chapter 3
Suppose is a linearly independent set in , and that is not in .
(a)
Is in
Suppose is in . Then we can write
Now consider two cases separately: either or . In either case, arrive at a
contradiction and conclude that is not in .
(b)
Is linearly independent?
If you assume linear dependence, you should be able to show is in the span of
the original set, which is a contradiction.
Suppose , , and are the rows of the matrix . Then we can interpret the solution to
the system of equations as the intersection of three planes containing the origin.
Discuss what this intersection would look like geometrically if the reduced row
echelon form of is of the form:
(a)
(b)
(c)
Are there any other possibilities?
Show that for any matrix , has a row of zeros if and only if one of the rows of can
be expressed as a linear combination of the others.
Recall that a plane in has an equation of the form , where , , are the components of
a normal vector to the plane. Suppose three planes intersect in a line, as shown
below.
We can find the line of intersection by solving a system of equations. Suppose the
augmented matrix corresponds to this system.
(a)
Explain why will have a row of zeros.
(b)
Consider the normal vectors to the three planes. What geometric property of
these particular three normal vectors explains the row of zeros in the reduced
row-echelon form?
All three normal vectors lie in the same plane, so one of
the normal vectors (rows) must be a linear combination of the other
two.