MATH 219 Exam 2, Fall 2014
American University of Beirut · Linear Algebra I · 10 pages · Prof. Kamal Makdisi
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Math 219 — Fall 2014
Quiz II, October 30. TIME: 75 minutes
GRADES (each problem is worth 14 points):
1
2
3
4
5
6
TOTAL/84
This exam booklet is also your answer sheet; please answer question 0 on this page,
and answer questions 1–6 inside the booklet. The problems are repeated inside the booklet.
There are extra blank sheets at the end for solutions. You can also use the back of any page for
solutions or scratchwork. If you need to continue a problem beyond the first page where it appears,
please INDICATE CLEARLY where the solution continues.
Read through the problems before starting, and decide which of them you wish to work on
first. Do as much of the exam as you can, and budget your time wisely. Each of questions 1–6
counts for 14 points. The exam is closed book, and calculators are not allowed.
Make sure to communicate your ideas by explaining all of your steps precisely and clearly.
Good luck!
0.
a) Your name:
b) Your AUB ID#:
1.
Consider the linear transformation T : R4 → R3 given by the matrix
AT =
1
2
1
2
2
4
3
0
4
8
5
4
.
(7 pts) a) Find a basis for ker T.
(7 pts) b) Show that T is not surjective.
2.
(7 pts) a) Show that {
2
1
9
,
2
0
1
,
2
0
2
} is a basis for R3.
(7 pts) b) Deduce in an easy way the dimension only dim ker T of the kernel ker T, where
the linear transformation T : R5 → R3 is given by the matrix
AT =
2
2
2
3
1
1
0
0
1
5
9
1
2
4
9
.
Do NOT find a basis for ker T!
3.
Let V be a vector space, and let ⃗x, ⃗y, ⃗z ∈ V . Let ⃗a = ⃗x, ⃗b = ⃗y + 2⃗x, and ⃗c = ⃗z − 4⃗x.
(7 pts) a) Show that Span{⃗a,⃗b,⃗c} = Span{⃗x, ⃗y, ⃗z}.
(7 pts) b) Show that {⃗x, ⃗y,⃗z} is a basis for V if and only if {⃗a,⃗b,⃗c} is a basis for V .
4.
(14 pts) Let V be a vector space with dim V = n. Let W ⊂ V be a subspace with dim W = n−1.
Show that there exists a linear transformation T : V → R such that ker T = W.
5.
Define a linear transformation T : P2 → P2 by T(f) = (x2 − 2)f ′′ − 2xf ′ + 2f.
(4 pts) a) Find T(1), T(x), and T(x2). (Check: T(x2) = −4.)
(10 pts) b) Find a basis for each of Image T and ker T.
6.
In R3, let ⃗w1 =
1
2
0
and ⃗w2 =
2
−1
1
. Let W = Span{⃗w1, ⃗w2}.
(4 pts) a) Show that ⃗w1 and ⃗w2 are orthogonal.
(6 pts) b) As usual, let ⃗e1 =
1
0
0
. Find the projections ProjW⃗e1 and ProjW ⊥⃗e1.
(4 pts) c) Extend {⃗w1, ⃗w2} to an orthogonal basis {⃗w1, ⃗w2, ⃗w3} of R3. This means that you
should find a possible vector for ⃗w3. Make sure to justify your reasoning.
0
1.
Consider the linear transformation T : R4 → R3 given by the matrix
AT =
1
2
1
2
2
4
3
0
4
8
5
4
.
(7 pts) a) Find a basis for ker T.
(7 pts) b) Show that T is not surjective.
1
2.
(7 pts) a) Show that {
2
1
9
,
2
0
1
,
2
0
2
} is a basis for R3.
(7 pts) b) Deduce in an easy way the dimension only dim ker T of the kernel ker T, where
the linear transformation T : R5 → R3 is given by the matrix
AT =
2
2
2
3
1
1
0
0
1
5
9
1
2
4
9
.
Do NOT find a basis for ker T! (Unless you are stuck and cannot find any other way.)
2
3.
Let V be a vector space, and let ⃗x, ⃗y, ⃗z ∈ V . Let ⃗a = ⃗x, ⃗b = ⃗y + 2⃗x, and ⃗c = ⃗z − 4⃗x.
(7 pts) a) Show that Span{⃗a,⃗b,⃗c} = Span{⃗x, ⃗y, ⃗z}.
(7 pts) b) Show that {⃗x, ⃗y, ⃗z} is a basis for V if and only if {⃗a,⃗b,⃗c} is a basis for V .
3
4.
(14 pts) Let V be a vector space with dim V = n. Let W ⊂ V be a subspace with dim W = n−1.
Show that there exists a linear transformation T : V → R such that ker T = W.
4
5.
Define a linear transformation T : P2 → P2 by T(f) = (x2 − 2)f ′′ − 2xf ′ + 2f.
(4 pts) a) Find T(1), T(x), and T(x2). (Check: T(x2) = −4.)
(10 pts) b) Find a basis for each of Image T and ker T.
5
6.
In R3, let ⃗w1 =
1
2
0
and ⃗w2 =
2
−1
1
. Let W = Span{⃗w1, ⃗w2}.
(4 pts) a) Show that ⃗w1 and ⃗w2 are orthogonal.
(6 pts) b) As usual, let ⃗e1 =
1
0
0
. Find the projections ProjW⃗e1 and ProjW ⊥⃗e1.
(4 pts) c) Extend {⃗w1, ⃗w2} to an orthogonal basis {⃗w1, ⃗w2, ⃗w3} of R3. This means that you
should find a possible vector for ⃗w3. Make sure to justify your reasoning.
6
Blank sheet.
7
Blank sheet.
8
Blank sheet.
9