This preview shows page 1. Sign up to view the full content.
BASIS
Math 21b, O. Knill
HOMEWORK: Section 3.2: 6,18,24,28,48,36*,38*
LINEAR SUBSPACE. A subset
X
of
R
n
which is closed under addition and scalar multiplication is called a
linear subspace
of
R
n
.
WHICH OF THE FOLLOWING SETS ARE LINEAR SPACES?
a) The kernel of a linear map.
b) The image of a linear map.
c) The upper half plane.
d) the line
x
+
y
= 0.
e) The plane
x
+
y
+
z
= 1.
f) The unit circle.
BASIS. A set of vectors
~v
1
, . . . , ~v
m
is a
basis
of a linear subspace
X
of
R
n
if they are
linear independent
and if they
span
the space
X
. Linear independent means that
there are no nontrivial
linear relations
a
i
~v
1
+
. . .
+
a
m
~v
m
= 0. Spanning the space
means that very vector
~v
can be written as a linear combination
~v
=
a
1
~v
1
+
. . .
+
a
m
~v
m
of basis vectors.
A
linear subspace
is a set containing
{
~
0
}
which is closed under
addition and scaling.
EXAMPLE 1) The vectors
~v
1
=
1
1
0
, ~v
2
=
0
1
1
, ~v
3
=
1
0
1
form a basis in the three dimensional space.
If
~v
=
4
3
5
, then
~v
=
~v
1
+ 2
~v
2
+ 3
~v
3
and this representation is unique. We can ±nd the coe²cients by solving
A~x
=
~v
, where
A
has the
v
i
as column vectors. In our case,
A
=
1
0
1
1
1
0
0
1
1
x
y
z
=
4
3
5
had the unique
solution
x
= 1
, y
= 2
, z
= 3 leading to
~v
=
~v
1
+ 2
~v
2
+ 3
~v
3
.
EXAMPLE 2) Two nonzero vectors in the plane which are not parallel form a basis.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.
 Spring '03
 JUDSON
 Linear Algebra, Algebra, Vectors

Click to edit the document details