Chapter 4
•Euclidean n-Space
•Linear Transformations from
Rn
to
Rm
•Properties of Linear Transformations
Rn
•Linear Transformations and Polynomials
to
Rm
Definition
If n is a positive integer, then an ordered n-tuple is a sequence of n
real numbers (a1,a2,…,an). The set of all ordered n-tuple is called n-space
and is denoted by R n.
Note that an ordered n-tuple (a1,a2,…,an) can be viewed either as a
“generalized point” or as a “generalized vector”
Definition
Two vectors u = (u1 ,u2 ,…,un) and v = (v1 ,v2 ,…, vn) in
equal if
u1 = v1 ,u2 = v2 , …, un = vn
Rn
are called
The sum u + v is defined by
u + v = (u1+v1 , u1+v1 , …, un+vn)
and if k is any scalar, the scalar multiple ku is defined by
ku = (ku1 ,ku2 ,…,kun)
Remarks
The operations of addition and scalar multiplication in this definition are
called the standard operations on R n .
The zero vector in
0 = (0, 0, …, 0).
Rn
is denoted by 0 and is defined to be the vector
If u = (u1 ,u2 ,…,un) is any vector in R n , then the negative (or additive
inverse) of u is denoted by -u and is defined by
-u = (-u1 ,-u2 ,…,-un).
The difference of vectors in
Rn
is defined by
v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)
Theorem 4.1.1 (Properties of Vector in R n )
If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn), and w = (w1 ,w2 ,…,wn) are vectors in
and k and m are scalars, then:
a) u + v = v + u
b) u + (v + w) = (u + v) + w
c) u + 0 = 0 + u = u
d) u + (-u) = 0; that is, u – u = 0
e) k(mu) = (km)u
f)
k(u + v) = ku + kv
g) (k+m)u = ku+mu
h) 1u = u
Euclidean Inner Product
Definition
If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn) are vectors in
Euclidean inner product u · v is defined by
Rn
, then the
u · v = u1 v1 + u2 v2 +… + un vn
Example
The Euclidean inner product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in
R4 is
u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18
It is common to refer to R n , with the operations of addition, scalar multiplication,
and the Euclidean inner product, as Euclidean n-space.
Theorem 4.1.2
If u, v and w are vectors in
a)
Rn
and k is any scalar, then
u·v=v·u
b) (u + v) · w = u · w + v · w
c) (k u) · v = k(u · v)
d)
v · v ≥ 0; Further, v · v = 0 if and only if v = 0
Example
(3u + 2v) · (4u + v)
= (3u) · (4u + v) + (2v) · (4u + v )
= (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v
=12(u · u) + 11(u · v) + 2(v · v)
Norm and Distance in Euclidean n-Space
We define the Euclidean norm (or Euclidean length) of a vector
u = (u1 ,u2 ,…,un) in R n by
|| u || (u  u)1/2  u12  u22 
 un2
Similarly, the Euclidean distance between the points u = (u1 ,u2 ,…,un) and
n
v = (v1 , v2 ,…,vn) in R defined by
d (u, v) || u  v || (u1  v1 )2  (u2  v2 )2 
 (un  vn )2
Example
If u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space
R4
|| u || 12  32  (2) 2  7 2  63  3 7
d (u, v) || u  v || (1  0) 2  (3  7) 2  (2  2) 2  (7  2) 2  58
Theorem 4.1.3 (Cauchy-Schwarz Inequality in R n )
If u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) are vectors in
|u · v| ≤ || u || || v ||
Theorem 4.1.4 (Properties of Length in R n )
If u and v are vectors in R n and k is any scalar, then
a) || u || ≥ 0
b) || u || = 0 if and only if u = 0
c) || ku || = | k ||| u ||
d) || u + v || ≤ || u || + || v || (Triangle inequality)
Rn
, then
Theorem 4.1.5 (Properties of Distance in Rn)
n
If u, v, and w are vectors in R and k is any scalar, then
a) d(u, v) ≥ 0
b) d(u, v) = 0 if and only if u = v
c) d(u, v) = d(v, u)
d) d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality)
Theorem 4.1.6
If u, v, and w are vectors in
Rn
with the Euclidean inner product, then
1
1
2
u  v  || u  v ||  || u  v ||2
4
4
Orthogonality
Definition
Two vectors u and v in
Rn are called orthogonal if u · v = 0
Example
4
In the Euclidean space R the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1)
are orthogonal, since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0
Theorem 4.1.7 (Pythagorean Theorem in R n )
If u and v are orthogonal vectors in R n whith the Euclidean inner
product, then
|| u  v ||2 || u ||2  || v ||2
Alternative Notations for Vectors in
Rn
It is often useful to write a vector u = (u1 ,u2 ,…,un) in
Row matrix or a column matrix:
u1 
u 
u   2
 
 
un 
or
u  [u1 u2
Rn
un ]
in matrix notation as a
Matrix Formulae for the Dot Product
If we use column matrix notation for the vectors
u1 
u 
u   2
 
 
un 
then
u  v  vT  u
Au  v  u  AT v
u  Av  AT u  v
and
 v1 
v 
v   2
 
 
 vn 
A Dot Product View of Matrix Multiplication
Let A = [aij] be an m×r matrix and B =[bij] be an r×n matrix, if the row vectors of
A are r1, r2, …, rm and the column vectors of B are c1, c2, …, cn , then the
matrix product AB can be expressed as
 r1  c1 r1  c2
r  c r  c
AB   2 1 2 2


 rm  c1 rm  c1
r1  cn 
r2  cn 


rm  c1 
In particular, a linear system Ax=b an be expressed in dot product form as
 r1  x  b1 
 r  x  b 
 2  2

  

  
r

x
 m  bm 
Where r1, r2,…, rm are the row vectors of A, and b1, b2, …, bm are the entries of b
Example:
3x1  4 x2  x3  1
2 x1  7 x2  4 x3  5
x1  5 x2  8 x3  0
(3, 4,1)  ( x1 , x2 , x3 )  1 

 
(2,

7,

4)

(
x
,
x
,
x
)
1
2
3  5 

(1,5, 8)  ( x1 , x2 , x3 )  0