1.
Linear dependence
A Denition of linear independence.
Pn
n
n
• v depends on S if there are {si }i=1 ⊂ S and {ai }i=1 ⊂ R such that v = i=1 ai si . P
n
n
n
• v depends on S if there are {si }i=1 ⊂ S and {ai }i=1 ⊂ R not all zero such that
Pnv = i=1 ai si .
n
n
• If v depends on S then there are {si }i=1 ⊂ S and {ai }i=1 ⊂ R such that v = i=1 ai si .
0
1
2
B Linear dependence of the zero vector. The problem is to decide if
depends on
,
0
0
0
in R2 .
1
2
0
a
+b
=
0
0
0
a + 2b
=
0
a
=
−2b
b
=
0
a
=
0
dependent
Suppose
thatthere
were
a, bsuch that
1
2
0
a
+b
=
0
0
0
Dependencewould
of a, b
follow
from
the
existence
1
2
0
such that a
+b
=
0
0
0 a + 2b
0
This is equivalent to
=
,
0
0
hence to a + 2b = 0
And hence equivalent to the existence of a, b such
that a = −2b.
Choosing b = 0, a = 0 this equality holds so such a, b
do exist.
Then a + 2b = 0
Thus a = −2b.
Thus if there were a, b and b = 0 then
a = 0 also.
2.
Linear maps on Rn
A Linearity in a concrete example.
Let T : R2 → R3 be a linear map, and suppose that T
3
1
= u and T
5
4
= v . Find an explicit vector
x ∈ R2 such that T x = 2u − 3v .
3
5
3
5
−9
• T x = 2u − 3v = 2T
− 3T
=T 2
−3
=T
. Therefore T x =
1 1
4
10
4
−9
−9
T
. Therefore x =
.
10
10
3
5
3
5
−9
−9
−3
=T
so T
= 2u−3v .
• 2u−3v = 2T
−3T
=T 2
1
4
1
4
−10
−10
B Linearity of a linear functional. OK
C Denition of Kernel. OK
D Basis.






5
3
5a + 3b
. Then ϕv = 2(5a +
2a
• Suppse v ∈ Ker ϕ and that v = a  2  + b  0  = 
0
−2
−2b
3b)
−
5(2a)
+
3(−2b)
=
0
.
Thus
any
vector
v
∈
Ker
ϕ
can
be
written
as a linear combination of
  

5
3
 2  ,  0  so they span Ker ϕ.
0
−2
1
3.
Linear maps on RR
For (Ta f )(x) = f (x + a), W = Span {erx | r ∈ R}.
B Image of a linear map.
Show that Ta W = W .
• Ta (er ) = er (a)er ∈ W so Ta W = W .
2