MAC 2103
Module 6
Euclidean Vector Spaces I
1
Learning Objectives
Upon completing this module, you should be able to:
1. Use vector notation in ℜn.
2. Find the inner product of two vectors in ℜn.
3. Find the norm of a vector and the distance between
two vectors in ℜn.
4. Express a linear system in ℜn in dot product form.
5. Find the standard matrix of a linear transformation
from ℜn to ℜm .
6. Use linear transformations such as reflections,
projections, and rotations.
7. Use the composition of two or more linear
transformations .
Rev.F09
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2
Euclidean Vector Spaces I
There are two major topics in this module:
Euclidean n-Space, ℜn
Linear Transformations from ℜn to ℜm
Rev.09
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3
Some Important
Properties of Vector Operations in ℜn
If u, v, and w are vectors in ℜn and k and s are scalars, then
the following hold: (See Theorem 4.1.1)
a) u + v = v + u
b) u + ( v + w ) = (u + v) + w
c) u + 0 = 0 + u = u
d) u + (-u) = 0
e) k(su) =(ks)u
f) k(u + v) = ku + kv
g) (k + s)u = ku + su
h) 1u = u
Rev.F09
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4
Basic Vector Operations in ℜn
Two vectors u = (u1, u2, … , un ) and v = (v1, v2,… , vn )
are equal if and only if
u1 = v1 , u2 = v2 , … , un = vn .
Thus,
u + v = (u1 + v1 , u2 + v2,…, un + vn)
u - v = (u1 - v1 , u2 - v2,…, un - vn)
and
5v - 2u = (5u1 - 2v1, 5u2 - 2v2,…, 5un - 2vn)
Rev.F09
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5
How to Find the Inner Product of
Two Vectors in ℜn ?
• The inner product of two vectors u = (u1,u2,…,un) and v =
(v1,v2,…,vn), u · v, in ℜn is also known as the Euclidean inner
product or dot product.
• The inner product, u · v, can be computed as follows:
r
rT r
u·v  u1  v1  u2  v2  ...  un  vn  u v
Example: Find the Euclidean inner product of u and v in ℜ4 , if
u = (2, -3, 6, 1) and v = (1, 9, -2, 4).
r
Solution:
u·v  (2)(1)  (3)(9)  (6)(2)  (1)(4)  33
Rev.F09


  2 3 6 1  


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
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1
9
2
4






6
How to Find the Norm of a Vector in ℜn ?
As we have learned in a previous module, the norm of a vector
in ℜ2 and ℜ3 can be obtained by taking the square root of the
sum of square of the components as follows:
 u1 
r

u  u  u , u  (u1 ,u2 )  
 u2 
2
1
2
2
 u1 


r
2
2
2 r
u  u1  u 2  u3 , u  (u1 ,u2 ,u3 )   u2 
 u 
 3 
Rev.F09
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7
How to Find the Norm of a Vector in ℜn? (Cont.)
Similarly, the Euclidean norm of u = (u1,u2,…,un), ||u||, in ℜn can
be computed as follows:
Example: Find the Euclidean norm of u = (2, -3, 6, 1) in ℜ4.
Solution:
r
u  u  u  ...  u , u  (u1 ,u2 ,...,un )
2
1
2
2
2
n
u  2 2  (3)2  6 2  12  4  9  36  1  50  5 2
Rev.F09
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8
How to Find the Distance Between
Two Vectors in ℜn ?
• The distance between u = (u1,u2,…,un) and v = (v1,v2,…,vn)
in ℜn , d(u,v), is also known as the Euclidean distance.
• The Euclidean distance, d(u,v), can be computed as
follows:
r r
r r
d(u, v)  u  v  (u1  v1 )2  (u2  v2 )2  ...  (un  vn )2
Example: Suppose u = (2, -3, 6, 1) and v = (1, 9, -2, 4).
Find the Euclidean distance between u and v in ℜ4 ,
Solution:
r r
r r
d(u, v)  u  v  (2  1)2  ((3)  9)2  (6  (2))2  (1  4)2
 1  144  64  9  218
Rev.F09
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9
How to Express a Linear System
in ℜn in Dot Product Form?
Example: Express the following linear system in dot product
form.
x1  5x2  2x3  9x4  2
3x1  2x2  6x3  2x4  5
4x1  x2  2x4  1
8x1  1x2  3x3  7x4  0
Solution:







(1, 5, 2, 9)  (x1 , x2 , x3 , x4 )  

(3, 2, 6, 2)  (x1 , x2 , x3 , x4 )  


(4,1, 0, 2)  (x1 , x2 , x3 , x4 )
 

(8, 1, 3, 7)  (x1 , x2 , x3 , x4 )  

Rev.F09
2
5
1
0


r
r r
  ai  x   b,i  1, 2, 3, 4



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10
How to Express a Linear Transformation
from ℜ3 to ℜ4 in Matrix Form?
The linear transformation T: ℜ3 → ℜ4 defined by the
equations w1  2x1  x2  x3
w2  x1  8x2  3x3
w3  x1  2x2  2x3
w4  6x1  x2  2x3
can be expressed in matrix form as follows:







w1  

w2  


w3
 

w4  

Rev.F09
2
1
1
6
1
8
2
1
1
3
2
2
 x 
 1 
r
r
r r r r
  x2   w  Ax, wi  aiT x  ai  x,i  1, 2, 3, 4


  x3 

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11
What is the Standard Matrix for
a Linear Transformation?
Based on our example in previous slide, the standard matrix
can be found from the linear transformation T: ℜ3 → ℜ4
expressed in matrix form.







w1  

w2  


w3
 

w4  

2
1
1
6
1
8
2
1
1
3
2
2
The standard matrix for T is:
Rev.F09
 x 
 1 
r
r
  x2   w  Ax


x
 3 



A



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2
1
1
6
1
8
2
1
1
3
2
2






12
Example and Notations
Example: Find the standard matrix for the
linear transformation T defined by the formula
as follows:
T (x1, x2 )  (3x1  2x2 , 7x1  x2 )  (w1,w2 )
Solution: In this case, the linear operator T
assigns a unique point (w1, w2) in ℜ2 to each
point (x1, x2) in ℜ2 according to the rule
Note:
A linear
transformation
T: ℜn → ℜm
is also known
as a linear
operator.
(w1,w2 )  (3x1  2x2 , 7x1  x2 ),
or as a linear system, it is as follows:
w1  3x1  2x2
w2  7x1  x2
Rev.F09
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13
Example and Notations (Cont.)
A linear system can be expressed in matrix form.
 w1   3


 w2   7
r
r
r
r
r
2   x1 



w

A
x

T
(
x)

T
(
x)

[T
]
x

A
1   x2 


In this case, the standard matrix for T is
 3
[T ]  [TA ]  A  
 7
2 

1 
In general, the linear transformation is represented by T: ℜn → ℜm or
TA: ℜn → ℜm; the matrix A = [aij] is called the standard matrix for the
linear transformation, and T is called multiplication by A.
Rev.F09
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14
Zero Transformation and Identity Operator
If 0 is the m x n zero matrix, then for every vector x in ℜn, we
will have the zero transformation from ℜn to ℜm, T0: ℜn →
ℜm, where T0 is called multiplication by 0.
r
r r
T0 ( x)  0 x  0
If I is the n x n identity matrix, then for every vector x in ℜn,
we will have an identity operator on ℜn , TI: ℜn → ℜn, where
TI is called multiplication by I.
r
r r
TI ( x)  Ix  x
Next, we will look at some important operators on ℜ2 and ℜ3, namely
the linear operators that produce reflections, projections, and
rotations.
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15
Linear Operators for Reflection
If the linear operator T: ℜ2 → ℜ2 maps each vector into its
symmetric image about the y-axis, we can construct a
reflection operator or linear transformation as follows:
r
r
T (u)  w  (w1,w2 )  (x  0y,0x  y)
w1  x  0y  w1  x
w2  0x  y  w2  y
y
(-x,y)
 w1   1 0   x 


 y 
w
0
1
 2  

 
(x,y)
w
u
x
Rev.F09
 1 0 
[T ]  

0
1


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16
Linear Operators for Reflection (Cont.)
If the linear operator T: ℜ2 → ℜ2 maps each vector into its
symmetric image about the x-axis, we can construct a
reflection operator or linear transformation as follows:
r
r
T (u)  w  (w1,w2 )  (x  0y,0x  y)
w1  x  0y  w1  x
w2  0x  y  w2  y
y
(x,y)
u
x
w
(x,-y)
Rev.F09
 w1   1 0   x 


 y 
 w2   0 1  

 1 0 
[T ]  

0
1


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17
Linear Operators for Reflection (Cont.)
If the linear operator T: ℜ2 → ℜ2 maps each vector into its
symmetric image about the line y = x, we can construct a
reflection operator or linear transformation as follows:
r
r
T (u)  w  (w1,w2 )  (0x  y, x  0y)
y
(y,x)
y=x
w
u
(x,y)
x
w1  0x  y  w1  y
w2  x  0y  w2  x
 w1   0 1   x 


 y 
 w2   1 0  

 0 1 
[T ]  

1
0


Rev.F09
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18
Linear Operators for Reflection (Cont.)
If the linear operator T: ℜ3 → ℜ3 maps each vector into its
symmetric image about the xy-plane, we can construct a
reflection
r
r operator or linear transformation as follows:
T (u)  w  (w1,w2 ,w3 )  (x  0y  0z,0x  y  0z,0x  0y  z)
w1  x  0y  0z  w1  x
w2  0x  y  0z  w2  y
w3  0x  0y  z  w3  z
z
u
(x,y,z)
y
w
(x,y,-z)
x
Rev.F09
 w1   1 0 0   x 

 


 w2    0 1 0   y 
 w   0 0 1   z 


 3 
 1 0 0 
[T ]   0 1 0 


http://faculty.valenciacc.edu/ashaw/  0
0 1 

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19
Orthogonal Projection Operator
If the linear operator T: ℜ3 → ℜ3 maps each vector into its
orthogonal projection on the xy-plane, we can construct a
projection
r
r operator or linear transformation as follows:
T (u)  w  (w1,w2 ,w3 )  (x  0y  0z,0x  y  0z,0x  0y  0z)
w1  x  0y  0z  w1  x
w2  0x  y  0z  w2  y
w3  0x  0y  0z  w3  0
z
u
(x,y,z)
y
w
(x,y,0)
x
Rev.F09
 w1  
 x 

  1 0 0 

y
w

 2   0 1 0 

 w   0 0 0  z 


 3  
 1 0 0 
[T ]   0 1 0 


http://faculty.valenciacc.edu/ashaw/  0
0 0 

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20
Orthogonal Projection Operator (Cont.)
Example: Use matrix multiplication to find the orthogonal
projection of (-9,4,3) on the xy-plane.
From previous slide, the standard matrix for the linear
operator T mapping each vector into its orthogonal projection
 1 0 0 
on the xy-plane in ℜ3 is obtained:
[T ]   0 1 0 


 0 0 0 
So the orthogonal projection, w, of (-9,4,3) on the xy-plane
 w1  
is:

 


  1 0 0   9   9 
 w2    0 1 0   4    4 
 w   0 0 0  3   0 

 

 3  
Thus, T(-9,4,3) = (-9,4,0).
Rev.F09
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21
Linear Operators for Rotation
If the linear operator T: ℜ2 → ℜ2 rotates each vector
counterclockwise in ℜ2 through a fixed angle θ in ℜ2, we can
construct a rotation operator or linear transformation as
follows:
r
r
T (u)  w  (w1,w2 )  (r cos(   ),r sin(   ))
 (x cos( )  ysin( ), x sin( )  y cos( ))
y
(w1,w2)
w1  r cos(   )
w
 r cos( )cos( )  r sin( )sin( )
θ u
ɸ
(x,y)
x
Hint: Let r = ||u||=||w||, then use x = r
cos(ɸ), y = r sin(ɸ), w1=r cos(θ+ɸ),
w2= r sin(θ+ɸ), and trigonometry
identities.
Rev.F09
 x cos( )  y sin( )
w2  r sin(   )
 r sin( )cos( )  r cos( )sin( )
 x sin( )  y cos( )
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22
Linear Operators for Rotation (Cont.)
w1  r cos(   )  x cos( )  ysin( )
w2  r sin(   )  x sin( )  y cos( )
y
(w1,w2)
 w1   cos( )  sin( )


 w2   sin( ) cos( )
w
θ u
ɸ
(x,y)
x
Rev.F09
 cos( )  sin( )
[T ]  
 sin( ) cos( )
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


23
 x 


y

 
Linear Operators for Rotation (Cont.)
Example: Use matrix multiplication to find the image of the
vector (3,-4) when it is rotated through an angle, θ, of 30°.
Since the standard matrix for the linear operator T rotating
each vector through an angle of θ (counterclockwise) in ℜ2
has been obtained:
 3
1

 cos( )  sin( )   cos(30o)  sin(30o)   2  2

[T ]  

o
o

3
 sin( ) cos( )   sin(30 ) cos(30 )   1
 2
2
Rev.F09
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24






Linear Operators for Rotation (Cont.)
It follows that the image, w, of (3,-4) when it is rotated
through an angle of 30° (counterclockwise) in ℜ2 can be
found as:


w  


3
2
1
2

1 
3
1



3( )  4( )


2  3
2
2



 

4
3 
1
3
 
3( )  4( )
2 

2
2
  3 34 
 

2


  3 4 3 
 

 
2

Thus,
 3 3  4 3 4 3
T (3, 4)  
,
2
2 

Rev.F09
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25
Composition of Linear Transformations
If TA: ℜn → ℜk and TB: ℜk → ℜm are linear transformations,
then the application of TA followed by TB produces a
transformation from ℜn to ℜm; this transformation is called
the composition of TB with TA and is denoted by TB ○ TA .
The composition TB ○ TA is linear because
r
r
(TB oTA )( x)  TB (TA ( x))
r
r
 B(A( x))  (BA)( x)
Thus, TB ○ TA is multiplication by BA and can be expressed
as TB ○ TA = TBA .
Alternatively, we have
[TB ○ TA ] = [ TB ][ TA ] .
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Composition of Linear Transformations (Cont.)
Example: Find the standard matrix for the stated
composition of linear operators on ℜ2, if a rotation of π/2 is
followed by a reflection about the line y = x.
We know the standard matrix for the linear operator TA
rotating each vector through an angle of θ = π/2
(counterclockwise) in ℜ2 is as follows:



 cos( )  sin( )   cos( 2 )  sin( 2 )
[TA ]  

 sin( ) cos( )   sin(  ) cos(  )

2
2

Rev.F09
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
 
   0 1 
  1 0 


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Composition of Linear Transformations (Cont.)
We also know the standard matrix for the linear operator, TB,
reflecting each vector about the line y = x in ℜ2 is as follows:
 0 1 
[TB ]  

1
0


The composition we want is the linear operator T:
T = TB ○ TA (rotation followed by reflection).
Therefore, the standard matrix for T is
[T] = [TB ○ TA ] = [ TB ][ TA ] .
 0 1   0 1   1 0 
[T ]  



1
0
1
0
0
1


 

Note: This is the symmetric image about the x-axis matrix. See slide 17.
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What have we learned?
We have learned to:
1. Use vector notation in ℜn.
2. Find the inner product of two vectors in ℜn.
3. Find the norm of a vector and the distance between
two vectors in ℜn.
4. Express a linear system in ℜn in dot product form.
5. Find the standard matrix of a linear transformation from
ℜn to ℜm .
6. Use linear transformations such as reflections,
projections, and rotations.
7. Use the composition of two or more linear
transformations .
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29
Credit
Some of these slides have been adapted/modified in part/whole from the
following textbook:
• Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition
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