Linear Algebra Test 2
Hello, and welcome to Test 2. Please complete as many problems as you can.
Show work for partial credit. If you use technology for any matrix calculations;
please indicate the following details: which technology did you use, what did you
input, and what was the output? You may consult your ebook, your notes, your
homework, my lecture videos, our review problems and solutions, and general
references on linear algebra. You may not consult resources where specific
problems are submitted, and their solutions posted, and your work must be
entirely your own.
If you have any questions, please contact me immediately.
1. Let ?? = [ 6 ?1 ?1 ?1
14 2 ?2 3 17 7 ?2 9
] and ?? = [
1 ?3 7 2
]. Is ?? ? nul ???
2. Let ?? = [ 2 6 1 ?1 1 3 5 13 1 3 ?2 ?8
]. Find a basis for nul ??.
3. Let ?? = [ 1 5 3 0 1 0 ?2 1 2 1 1 5 3 0 1
]. Find a basis for col ??.
4. Let ?? = [
1 1 2 3 4 9 8 27
] and ?? = [
64 101 121 ?1
]. Is ?? ? col ???
5. If ?? is a 3 × 8 matrix with at least one non-zero entry, what are the possible dimensions of its null space?
6. Calculate the determinants using the indicated methods:
a) Expand along a convenient row or column, without using row or column
operations:
|
1 0 0 3 0 2 2 0 0 2 3 0 3 0 0 5
| =
b) Use row/column operations before expanding:
|
2 ?10 30 10 1 2 ?10 30 3 1 2 ?10
?1 3 1 2
| =
7. Use Cramer’s rule to find the value of ?? in the system:
{ 18?? + 3?? = 55
153?? + 49?? = 211
8. In ?4, let ? = {????, ????}, with ???? = [
1 2 4 8
] and ???? = [
1 3 9
27
], and let ?? be the
subspace that they span. What is the ?4 vector ?? ? ?? that has the
coordinate vector [??]? = [ 9
?4 ], with respect to this basis?
9. In ?2 the subset ?? = {[ ?? ??] |??
2 ? ??2 = 0} is not a subspace. Verify this, by
showing that it fails to satisfy one of the three required conditions.
10. Consider the vector space ?2 of real polynomial functions with degree not
exceeding 2, or more formally defined:
?2 = {??: ? ? ? | ??(??) = ??0 + ??1?? + ??2?? 2; ??0, ??1, ??2 ? ? },
and consider the transformation ??: ?2 ? ?2, where ??(??) is the function ??
given by the rule ??(??) = ??(??) ? ???? ?(0).
a) Evaluate ??(??) for the function given by ??(??) = 9 + 5?? 3?? 2.
b) Write down the standard matrix for the transformation ??, using the
basis {??1, ??2, ??3}, where ??1(??) = 1, ??2(??) = ??, and ??3(??) = ?? 2 You may
assume that ?? is linear without proving it (although, see bonus
question).
11. Let ??1 = [ 3 3
] , ??2 = [ ?2 2
] , ??1 = [ 2
?1 ] , ??2 = [
2 1
] , and note that both
? = {??1, ??2} and
?? = {??1, ??2}
are bases for ?2.
Write down the matrices ?? ???
and ?? ????
, and use them to obtain the ??
coordinate vector for the vector ?? with ? coordinate vector [??]? = [ 3
?4 ].
?? ???
=
?? ????
=
[??]?? =
