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Chapter 4
EUCLIDEAN VECTOR SPACES
Contents
4.1 Vectors in n-interval ................................................................................................................... 2
4.2 Linear Transformations From Rn To Rm ................................................................................ 8
4.3 Properties Of Linear Transformation ................................................................................... 22
1
4.1 Vectors in n-duration
DEFINITION
If n is a stubborn integer, then an order n-tuple is a arrangement of n real numbers (a1 , a2 , . . . , some)
The set of all ordered n-tuples is called n-extension and is denoted by Rn.
R1 - R sink of real numbers
R2 - ordered brace (a1 , a2)
R3 - ordered triple (a1 , a2 , a3 )
Rn. - ordered n-tuple (a1 , a2 , . . . , an)
The symbol (a1 , a2 , . . . , an) has pair different geometric interpretation that is for the re~on that a point or a
vector.
DEFINITION
1. Two vectors u = (u1 , u2 , . . . , un) and v = (v1 , v2 , . . . , vn) in Rn. are alike if u1 = v1
v 2 , . . . , un = v n .
2. The ~marize u + v is defined by
u + v = ( u1 + v 1 , u2 + v 2
,... ,
,
u2 =
un + v n )
3. If k is ~ one scalar, the scalar multiple ku is defined through ku = ( ku1 , ku2 , . . . , kun )
The naught vector in Rn is denoted by 0 : 0 = (0, 0, . . . ,0)
If u = (u1 , u2 , . . . , un) in a single one vector in Rn then the negative (additive inverse) of u denoted by –u
–u = (–u1 , –u2 , . . . , –un)
The amount of inequality of vector in Rn is defined ~ the agency of u v = u + (v) = ( u1 v1 , u2 v2
,... ,
un v n )
Properties of Vector Operations in nspace

Properties of Vector in Rn
If u = (u1 , u2 , . . . , un) , v = (v1 , v2 , . . . , vn) and
and m are scalars soon afterward
(a) (b) (u + v) + w = u + (v + w)
(c)
2
u+v=v+u
u+0=0+u
w = (w1 , w2 , . . . , wn) are vectors in Rn and k
(d) u + (u ) = 0
u–u=0
(e) k( m u ) = ( k m ) u
(f) k (u + v) = ku + kv
(g) (k + m) u = ku + mu
(h) 1u=u
Example 1
Let u = (2 , 1, 3, 4 ) , v = ( 3, 2, 0, 4), w = ( 0,...

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