Math 375 – Midterm Exam II
October 27, 2005
Note:
There are
five
problems. Do as much as you can during this exam. Take
this exam sheet home. If you do not have time to finish all the problems you should
complete the exam at home and hand in what you did for partial credit (this is due
on Monday in class).
1.
(i) (10pts.)
Let
V
be the vector space of all vectors in
R
3
which are linear
combinations of
1
1
1
and
1
0
0
. Find an orthonormal basis of
V
.
(ii) (15 pts) Let
V
be as in (i) and let
a
=
1
1

2
.
Find the vector
b
in
V
which minimizes the distance
x

a
among all vectors
x
in
V
; i.e. find
b
so that
b

a
= min
x
∈
V
x

a .
Compute this minimal distance.
2.
(30 pts) Consider the linear transformation
R
3
→
R
4
given by
T
(
x
) =
x
1

x
2
x
2

x
3
x
1

x
3
0
(i) Find a basis of the nullspace of
T
.
(ii) Find a basis of the range of
T
.
(iii) For each
b
∈
R
4
give a complete description of the set of solutions of the
linear system
T
(
x
) =
b.
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 Fall '08
 Staff
 Math, Calculus, Linear Algebra, Algebra, 10pts, 15 pts, rn rn, Rn span Rn, linear transformation R3

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