Page 62 - Advanced Linear Algebra

46 Advanced Linear Algebra

and
# ~ bÄb # ~ ! b Äb !

where the 's are distinct, the 's are distinct and all coefficients are nonzero,

!
then ~ and after a reindexing of the ! 's if necessary, we have ~ and

for all ~ ÁÃÁ . Note that this is stronger than saying that
(
~ !
~ ! .)

We may characterize linear independence as follows.
Theorem 1.6 Let : £ ¸ ¹ be a nonempty set of vectors in . The following are
=
equivalent:
)
1 : is linearly independent.
)
2 Every nonzero vector # span²:³ is an essentially unique linear
combination of the vectors in .
:
3 No vector in is a linear combination of other vectors in .
)
:
:
Proof. Suppose that 1 holds and that
)
£ # ~ bÄb ~ ! bÄb !

where the 's are distinct, the 's are distinct and the coefficients are nonzero.

!
By subtracting and grouping 's and 's that are equal, we can write

!

~ ² c ³ bÄb² c ³

b b b Äb
b

c ! b c Äc !
b
)
and so 1 implies that ~ ~ and ~ " " and ~ ! " " for all ~ Á Ã Á .
Thus, 1 implies 2 .
)
)
If 2) holds and : can be written as

~ bÄb

where  : are different from , then we may collect like terms on the right

and then remove all terms with coefficient. The resulting expression violates

2). Hence, 2) implies 3). If 3) holds and
bÄb ~

£ , then € and we may write
where the 's are distinct and

~ c ² bÄb ³

)
which violates 3 .
The following key theorem relates the notions of spanning set and linear
independence.