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University of Toronto at Scarborough
Midterm Test
MATA23H3
Linear Algebra I
Examiner: N. Cheredeko
Date: February 10, 2006
Duration: 110 minutes
1.
[10 points]
(a) Determine whether

1
3
2
is a linear combination of
2
0
1
and
0
2
4
.
(b) Do
2
0
1
and
0
2
4
span
R
3
? Justify your answer.
2.
[10 points]
(a) Find the projection of
w
=
1
1
1
on the normal to the plane
π
with equation
x

7
y
+ 4
z
= 0.
(b) Find the distance from the point
P
(1
,
1
.
1) to the plane
π
.
3.
[15 points]
Let
u
=
3
4
0
and
v
=
0
0
3
.
(a) Find the angle between
u
and
v
.
(b) State and verify the CauchySchwartz inequality for
u
and
v
.
(c) Find a unit vector that is perpendicular to both
u
and
v
.
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page
2
4.
[15points]
(a) Suppose
A
and
B
are invertible matrices and
X
is a matrix such that
AXB
=
A
+
B .
What is
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