MATH 219 Final, Fall 2007
American University of Beirut · Linear Algebra I · 2 pages
More MATH 219 files
Show the text of this final exam
American University of Beirut
Final – Fall 2007-2008
Math 219
I – Let T:
3
3
R → R
defined by
+
−
−
+
=
3
1
2
3
2
1
3
2
1
7x
x
x
x
2x
x
x
x
x
T
.
a- Show that T is a linear transformation.
b- Write T as a matrix with respect to the standard basis of
3
R .
c- Find a basis for the kernel and image of T.
II- Let U:
2
3
R → R
and T:
3
3
R → R
be the linear transformation given by the matrices
representations:
−
−
−
−
= −
1
2
3
1
2
3
U
and
−
=
3
0
6
1
1
0
0
1
2
T
.
a- Find the matrix of UoT.
b- Show that
KerU
ageT
Im
6
0
2
∩
∈
−
III- Let
2
P be the vector space of all polynomials of degree less or equal than two and
2
2
P
T : P
→
be the linear transformation defined by:
2
2
0
2
1
2
1
0
2
2
1
0
2a )x
(a
8a )x
(a
2a )
6a
(5a
a x )
a x
T(a
T(p(x))
−
+
+
−
+
+
=
+
+
=
a- Find the eigenvalues of T with respect to the standard basis of
2
P .
b- Find bases for the eigenspaces of T.
IV- Let u and v be in
3
R . Define
3
3
2
2
1 1
3u v
2u v
u v
u, v
+
+
>=
<
.
a- Show that
3
3
2
2
1 1
3u v
2u v
u v
u, v
+
+
>=
<
is an inner product
b- Use the Gram-Schmidt process to transform
u =( ,1 0,0)
,
0)
,1,1
(
v =
and
)
1,1,1
w =(
into an orthonormal basis.
V- Define
2
2
1
R
u ,u
∈
such that
=
=
2
1
1 ,u
1
u
2
1
.
a- Show that
1 u2
u ,
span
2
R .
b- Let B = {
1 e2
e ,
} be the standard basis for
2
R . Write
1 e2
e ,
as a linear
combination
of
1 u2
u ,
.
c- Let T:
3
2
R
R
→
with
B
T its matrix representation . And let
=
9
1
2
(u )
T
1
B
and
=
5
2
3
(u )
T
2
B
. Determine
B
T .
VI- True or False. If it is True explain fully. If it is false give a counter-example.
a- If A is a 3x4 matrix then the system Ax = 0 has at least one independent variable.
b- T:
V → W
is a linear transformation . Let
V
v , v ...v
n
2
1
∈
, then if
T(v ),T(v )...T(v )
n
2
1
are linearly independent then
n
2
1
v , v ...v
are linearly
independent.
c-
VII- Prove the following independent statements:
a- An nxn matrix is said to be idempotent if
A 2 = A
. Show that if λ is an eigenvalue
of an idempotent matrix thenλ = 0 orλ = 1.
b- Let
3
P be the vector space of all polynomials of degree less or equal than three.
For any polynomial p(x) we define p’(x) to be the derivative of p(x). Let
3
2
4x
x
x
1
g(x)
+
−
= +
. Show that C = {g(x), g’(x), g’’(x), g’’’(x)} is a basis for
3
P .
c- Let A be a diagonalizable matrix whose eigenvalues are all 1 or -1. Show that A is
invertible and that
A 2 = I
.
d- Let B be an n x n invertible matrix. Prove that
n n
n n
M
T : M
×
× →
A
AB
T(A)
=
→
is an isomorphism.