CO350 Assignment 2
–
Wing Man Fok
20300650
,
May 21 2010
:

Instructor Mathieu Guay Paquet
( )
1 a
[
]
,…,
+
,
contrapositive Suppose the set a1
ak 1 is linearly dependent
,
,…,
+
+…
Then there exists scalars c1
ck 1 not all zero such that c1a1
+
+
+ = …(*)
ck 1ak 1 0
+ ≠ .
,
,…,
,
+…
Note that ck 1 0 Otherwise since a1
ak is linearly independent c1a1
+
+ =
=
∈ ,…, ,
.
ckak 0 0 implies ci 0 for all i 1
n which is impossible
*,
Continue from
+ =
+ 
+…+
= =

+ 
ak 1
ck 1 1c1a1
ckak i 1k ck 1 1ciai
+
,…,
,
+ ∈
,…,
Since ak 1 is a linear combination of a1
ak ak 1 spana1
ak
.
This completes the proof
( )
1 b
,…,
.
It suffices to show that a1
akcan be extended to a basis for Rm First check
≥ .
={
,…,
}
= .
> ,
+ ∉
∪ .
,
that m k
Let A
a1
ak and B ϕ When m k choose ak 1 spanA B Then
+
,
( )
∪
.
add the vector ak 1 to the set B by part a A B is linearly independent For
,
,
∪
finite dimensional spaces in this case Rm we can repeat this process until A B