Page 63 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

General Properties of Vector Spaces Notes
Let V be a vector space over a field F then
1. a O = O for a F, O V
2. O = O for O F, V

3. ( a ) ( )a (a ) for a F , V

,
4. ( a u ) au a for a F u and V
5. If a = 0 then either a = 0 or V = 0 for a F, V.
Proof:
1. L.H.S = a O
= a (O + O) (because O = O + O)

= a O + a O (distributive law)
Thus aO = aO + aO or aO + O = aO + aO
Hence by cancellation law we get
aO = O.

2. L.H.S. = O (O O ) (because O = 0 + 0)
= 0 0 (distributive law)

Thus 0 = 0 0
or 0 + 0 = 0 + 0
Hence by cancellation law
= 0 = 0.

3. a ( a ) = a ( ) a 0 0
Therefore a is additive inverse of a(– ).

Again a ( a ) ( )a Oa 0.
Therefore a is additive inverse of (– ) a.
i.e. (– v) a = – av

4. L.H.S. = a (u )
= a [u ( )]
= au a ( ) [by property (3)]

= a u – a
= R.H.S.
5. If a = 0 then the proposition is true.
–1
But if a 0 then a exists in F.
a = 0 a 1 (a ) a 1 0 (a 1 ) a
= 0 1 . 0 0.

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