Page 64 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Cancellation
Let V be a vector space over a field F, then
(i) a b a b for , b F and V , 0.
a
(ii) au a u for a F a 0, and , V .
u
Proof:
(i) L.H.S. = a b or a b 0

or (a b ) 0.

Since 0, therefore, we must have
a b 0 or a b
(ii) L.H.S. au a

or (a u ) 0

Since a 0, we must have
u 0 u

Example 34: Let F be a field and let V be the totality of all ordered n-tuples ( 1 , 2 ,.... n )
where 1 . F Two elements ( 1 , 2 ,.... n ) and ( 1 , 2 ,.... ) of V are declared to be equal if
n
and only if i i for each i = 1, 2, ..., n. We now introduce the requisite operations in V to make
of it a vector space by defining:

1. ( 1 , 2 ,.... n ) + ( 1 , 2 ,.... ) = ( 1 1 , 2 2 ,....., n n )
n
2. a ( 1 , 2 ,.... n ) = (a 1 , a 2 ,....a n ) for a F
It is easy to verify that with these operations, V is a vector space over F.

Example 35: Let F be any field and let V = F(x), the set of polynomials in x over F. We
merely choose the fact that two polynomials can be added to get again a polynomial and that a
polynomial can always be multiplied by an element of F. With these natural operations F(x) is
a vector space over F.

Example 36: The set of continuous real-valued functions on the real line is a real vector
space with addition of functions f + g and multiplication by real numbers as its laws of composition.

2
d y
Example 37: The set of solution of the differential equation 2 y is a real vector space.
dx
Self Assessment

,
17. Show that the set W of ordered tried ( ,a a 2 , 0) where a a 2 F is a vector space.
1
1
z
y
x
y
x
18. Prove that the set W ( ,2 ,4 ) : , ,z R is a vector space.
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