University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Term Test
MATB24H
Linear Algebra II
Examiner: S. Tryphonas
Date: November 4, 2006
Duration: 110 minutes
1.
[10 points]
Let
V
be a vector space over the field
F
. Let
z
be the zero vector of
V
and 0 be the zero element of
F
.
(a) Using
only
the conditions listed in the definition of a vector space, prove that for
each
v
∈
V
and
a
∈
F
:
i. (

a
)
v
=

(
a
v
)
.
ii.
a
(

v
) =

a
(
v
)
.
(b) Determine whether the set
V
of all 2 x 2 matrices with the standard matrix
addition and scalar multiplication defined by
r
A
=
r
·
A
T
for all
A
∈
V
and
r
∈
R
is a real vector space.
2.
[10 points]
(a) Give the definition of a subspace
W
of a real vector space
V
.
(b) Let
T
:
V
→
V
be a linear transformation from the vector space
V
into the
vector space
V
. If
W
is a subspace of
V
, prove that
T
[
W
], the image of
W
under
T
is a subspace of
V
3.
[10 points]
Let
V, V
be vector spaces over
R
.
Let
T
:
V
→
V
be a linear
transformation. Suppose
v
1
,
v
2
,
· · ·
,
v
k
∈
V
.
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 Fall '09
 YANG
 Linear Algebra, Algebra, Vector Space, linear transformation

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