MAC 2103

1
MAC 2103

• Module 6
• Euclidean Vector Spaces I

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Learning Objectives

• Upon completing this module, you should be able
  to
• Use vector notation in Rn.
• Find the inner product of two vectors in Rn.
• Find the norm of a vector and the distance
  between two vectors in Rn.
• Express a linear system in Rn in dot product
  form.
• Find the standard matrix of a linear
  transformation from Rn to Rm .
• Use linear transformations such as reflections,
  projections, and rotations.
• Use the composition of two or more linear
  transformations .

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Rev.F09
3
Euclidean Vector Spaces I
There are two major topics in this module
Euclidean n-Space, Rn Linear Transformations
from Rn to Rm
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Rev.09
4
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)
• 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

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Rev.F09
5
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)
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Rev.F09
6
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
• Example Find the Euclidean inner product of u
  and v in R4 , if u (2, -3, 6, 1) and v (1, 9,
  -2, 4).
• Solution

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Rev.F09
7
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

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Rev.F09
8
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

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Rev.F09
9
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
• Example Suppose u (2, -3, 6, 1) and v (1, 9,
  -2, 4).
• Find the Euclidean distance between u and v in R4
  ,
• Solution

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Rev.F09
10
How to Express a Linear System in Rn in Dot
Product Form?
Example Express the following linear system in
dot product form. Solution
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Rev.F09
11
How to Express a Linear Transformation from R3
to R4 in Matrix Form?
The linear transformation T R3 ? R4 defined by
the equations can be expressed in matrix form
as follows
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Rev.F09
12
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. The standard matrix for T is
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Rev.F09
13
Example and Notations
Example Find the standard matrix for the linear
transformation T defined by the formula as
follows 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 or as a linear system, it is as follows
Note A linear transformation T Rn ? Rm is
also known as a linear operator.
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Rev.F09
14
Example and Notations (Cont.)
A linear system can be expressed in matrix
form. In this case, the standard matrix for
T is
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.
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Rev.F09
15
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. 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.

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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Rev.F09
16
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 y
(-x,y) (x,y) w
u x
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Rev.F09
17
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
y
(x,y) u
x w
(x,-y)
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Rev.F09
18
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
y (y,x)
y x
w
u (x,y)
x
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Rev.F09
19
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 operator
or linear transformation as follows
z
u
(x,y,z) y
w (x,y,-z) x
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Rev.F09
20
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 operator
or linear transformation as follows
z
u
(x,y,z) y
w (x,y,0) x
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Rev.F09
21
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 on the
xy-plane in R3 is obtained So the orthogonal
projection, w, of (-9,4,3) on the xy-plane
is Thus, T(-9,4,3) (-9,4,0).

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Rev.F09
22
Linear Operators for Rotation
If the linear operator T R2 ? R2 rotates each
vector counterclockwise in R2 through a fixed
angle ? in R2, we can construct a rotation
operator or linear transformation as follows

y (w1,w2) w ? u
(x,y) ?
x
Hint Let r uw, then use x r
cos(?), y r sin(?), w1r cos(??), w2 r
sin(??), and trigonometry identities.
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Rev.F09
23
Linear Operators for Rotation (Cont.)

y (w1,w2) w
? u (x,y) ?
x
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Rev.F09
24
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, ?, of 30. Since the standard
matrix for the linear operator T rotating each
vector through an angle of ? (counterclockwise)
in R2 has been obtained
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Rev.F09
25
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 Thus,
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Rev.F09
26
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 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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Rev.F09
27
Composition of Linear Transformations (Cont.)
Example Find the standard matrix for the stated
composition of linear operators on R2, if a
rotation of p/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 ? p/2
(counterclockwise) in R2 is as follows
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Rev.F09
28
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 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
. Note This is the symmetric image about
the x-axis matrix. See slide 17.
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Rev.F09
29
What have we learned?

• We have learned to
• Use vector notation in Rn.
• Find the inner product of two vectors in Rn.
• Find the norm of a vector and the distance
  between two vectors in Rn.
• Express a linear system in Rn in dot product
  form.
• Find the standard matrix of a linear
  transformation from Rn to Rm .
• Use linear transformations such as reflections,
  projections, and rotations.
• Use the composition of two or more linear
  transformations .

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
Rev.F09
30
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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to download other modules.
Rev.F09