Section 6
Vector spaces and subspaces
Problem 1
Define the following concepts :
a)
S is a subspace
b)
The subspace S = span{
w
v
u
,
,
}
c)
Linear independence of vectors
w
v
u
,
,
.
d)
The vectors
w
v
u
,
,
are a basis for a subspace S.
Problem 2
Determine whether or not the following are subspaces.
a)
From R
3
, let S
a
be the set of all vectors whose length is 1.
b)
From R
2x2
, let S
b
be the set of all 2 x 2 matrices whose sum of all entries equal 0.
c)
From P
3
, let S
c
be the set of all 3
rd
degree polynomials such that p(0) = 1.
Solution
a)
not a subspace
b)
subspace
c)
not a subspace
Problem 3
Consider the matrices
1
1
1
1
1
0
3
1
2
1
0
1
3
2
1
M
M
M
a)
Show that the matrices
M
1
,
M
2
and
M
3
are linearly independent.
b)
Express the matrix
3
5
5
6
M
as a linear combination of
M
1
,
M
2
and
M
3
.
c)
Find a new set of linearly independent matrices
N
1
,
N
2
and
N
3
such that
span{
N
1
,
N
2
,
N
3
} = span{
M
1
,
M
2
,
M
3
}.
Solutions
a)
The only solution to
aM
1
+ bM
2
+ c M
3
= 0
is
a = 0, b = 0, c = 0.
b)
We want to obtain a, b, c such that
aM
1
+ bM
2
+ c M
3
= M
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1
0
1
3
1
1
6
5
1
2
0
1
1
1
5
3
3
6
5
2
5
3
a
b
c
a
b
c
b
c
a
c
a
b
c
Solving for a, b and c, we find a = 3, b = 1, c = 2
c)
Replace any matrix by a scalar multiple, or by a combination of two of them… or even
better reduce the basis vectors to their simplest form:
1
0
1
2
1
0
1
2
1
0
1
2
1
3
0
1
0
3
1
3
0
1
0
3
1
1
1
1
0
1
0
3
0
3
1
3
1
0
1
2
1
0
0
8
0
1
0
3
0
1
0
3
0
0
1
6
0
0
1
6
1
2
3
1
0
0
1
0
0
0
8
0
3
1
6
N
N
N
Problem 4
The set of all 2
nd
degree polynomials, P , is a vector space
under the following rules of addition
and scalar multiplication :
Let
p(
x
)
= a
0
+ a
1
x
+ a
2
x
2
and
q(
x
)
= b
0
+ b
1
x
+ b
2
x
2
(p + q)(
x
) = (a
0
+ b
0
)+ (a
1
+ b
1
)
x
+ (a
2
+ b
2
)
x
2
=
p(
x
) + q(
x
);
(kp)(
x
) = ka
0
+ ka
1
x
+ ka
2
x
2
= k[p(
x
)].
a)
Show that S
1
, the set of all elements p (
x
)
P such that p(2) = 0, is a subspace.
b)
Find a basis for the S
1
. State its dimension.
c)
Show that the polynomial p(
x
)
= 2 + 3
x
2
x
2
is an element of S
1
and express it as a
combination of the basis for S
1
.
d)
Show that S
2
, the set of all elements p (
x
)
P such that
p’(1) = 0, is a subspace.
e)
Find a basis for the S
2
. State its dimension.
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 Fall '15
 Linear Algebra, Vector Space, basis, Sc

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