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Linear Algebra
Chapter 4
Vector Spaces
4.1 The vector Space Rn
Definition 1.
Let (u1 , u2 , ..., un ) be a sequence of n real numbers. The set of all
such sequences is called n-space (or n-dimensional. space) and
is denoted Rn.
u1 is the first component of (u1 , u2 , ..., un ) .
u2 is the second component and so on.
Example 1
• R2 is the collection of all sets of two ordered real numbers.
For example, (0, 0) , (1, 2) and (-2, -3) are elements of R2.
• R3 is the collection of all sets of three ordered real numbers.
For example, (0,0, 0) and (-1,3, 4) are elements of R3.
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Definition 2.
Let u  (u1 , u2 , ..., un ) and v  (v1 , v2 , ..., vn ) be two elements of Rn.
We say that u and v are equal if u1 = v1, …, un = vn.
Thus two elements of Rn are equal if their corresponding
components are equal.
Definition 3.
Let u  (u1 , u2 , ..., un ) and v  (v1 , v2 , ..., vn ) be elements of Rn
and let c be a scalar. Addition and scalar multiplication are
performed as follows:
u  v  (u1  v1 , ..., un  vn )
Addition:
cu  (cu1 , ..., cun )
Scalar multiplication :
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► The set Rn with operations of componentwise addition and
scalar multiplication is an example of a vector space, and its
elements are called vectors.
We shall henceforth interpret Rn to be a vector space.
(We say that Rn is closed under addition and scalar multiplication).
► In general, if u and v are vectors in the same vector space, then u + v is
the diagonal of the parallelogram defined by u and v.
Figure 4.1
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Example 2
Let u = ( –1, 4, 3) and v = ( –2, –3, 1) be elements of R3.
Find u + v and 3u.
Solution:
Example 3
In R2 , consider the two elements
(4, 1) and (2, 3).
Find their sum and give a geometrical
interpretation of this sum.
we get (4, 1) + (2, 3) = (6, 4).
The vector (6, 4), the sum, is the
diagonal of the parallelogram.
Figure 4.2
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Example 4
Consider the scalar multiple of the vector (3, 2) by 2, we get
2(3, 2) = (6, 4)
Observe in Figure 4.3 that (6, 4) is a vector in the same direction
as (3, 2), and 2 times it in length.
Figure 4.3
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Zero Vector
The vector (0, 0, …, 0), having n zero components, is called the
zero vector of Rn and is denoted 0.
Negative Vector
The vector (–1)u is writing –u and is called the negative of u.
It is a vector having the same length (or magnitude) as u, but
lies in the opposite direction to u.
u
-u
Subtraction
Subtraction is performed on element of Rn by subtracting
corresponding components.
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Theorem 4.1
Let u, v, and w be vectors in Rn and let c and d be scalars.
(a) u + v = v + u
(b) u + (v + w) = (u + v) + w
(c) u + 0 = 0 + u = u
(d) u + (–u) = 0
(e) c(u + v) = cu + cv
(f) (c + d)u = cu + du
(g) c(du) = (cd)u
(h) 1u = u
Figure 4.4
Commutativity of vector addition
u+v=v+u
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Example 5
Let u = (2, 5, –3), v = ( –4, 1, 9), w = (4, 0, 2) in the vector space R3.
Determine the vector 2u – 3v + w.
Solution
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Column Vectors
Row vector: u  (u1 , u2 , ..., un )
Column vector:
 u1 

 
un 
We defined addition and scalar multiplication of column vectors
in Rn in a componentwise manner:
 u1   v1   u1  v1 
       
un  vn  un  vn 
    

and
 u1   cu1 
c      
un  cun 
   
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Homework
Exercise set 1.3 page 32:
3, 5, 7, 9.
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4.2 Dot Product, Norm, Angle, and Distance
Definition
Let u  (u1 , u2 , ..., un ) and v  (v1 , v2 , ..., vn ) be two vectors in Rn.
The dot product of u and v is denoted u.v and is defined
by
u  v  u1v1 .   un vn
The dot product assigns a real number to each pair of vectors.
Example 1
Find the dot product of
u = (1, –2, 4) and v = (3, 0, 2)
Solution
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Properties of the Dot Product
Let u, v, and w be vectors in Rn and let c be a scalar. Then
1. u.v = v.u
2. (u + v).w = u.w + v.w
3. cu.v = c(u.v) = u.cv
4. u.u  0, and u.u = 0 if and only if u = 0
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Norm of a Vector in Rn
Definition
The norm (length or magnitude)
of a vector u = (u1, …, un) in Rn
is denoted ||u|| and defined by
u 
u1 2    un 2
Note:
The norm of a vector can also
be written in terms of
the dot product u  u  u
Figure 4.5 length of u
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Example 2
Find the norm of each of the vectors u = (1, 3, 5) of R3
and v = (3, 0, 1, 4) of R4.
Solution
Definition
A unit vector is a vector whose norm is 1.
If v is a nonzero vector, then the vector
is a unit vector in the direction of v.
This procedure of constructing a unit vector in the same direction
as a given vector is called normalizing the vector.
1
u v
v
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Example 3
(a) Show that the vector (1, 0) is a unit vector.
(b) Find the norm of the vector (2, –1, 3). Normalize this vector.
Solution
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Angle between Vectors ( in R2)
The law of cosines gives:
← Figure 4.6
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Angle between Vectors (in R n)
Definition
Let u and v be two nonzero vectors in Rn.
The cosine of the angle  between these vectors is
uv
cos  
0  
u v
Example 4
Determine the angle between the vectors u = (1, 0, 0) and
v = (1, 0, 1) in R3.
Solution
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Orthogonal Vectors
Definition
Two nonzero vectors are orthogonal if the angle between them is a
right angle .
Theorem 4.2
Two nonzero vectors u and v are orthogonal if and only if u.v = 0.
Proof
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Example 5
Show that the following pairs of vectors are orthogonal.
(a) (1, 0) and (0, 1).
(b) (2, –3, 1) and (1, 2, 4).
Solution
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Note
(1, 0), (0,1) are orthogonal unit vectors in R2.
(1, 0, 0), (0, 1, 0), (0, 0, 1) are orthogonal unit
vectors in R3.
(1, 0, …, 0), (0, 1, 0, …, 0), …, (0, …, 0, 1) are
orthogonal unit vectors in Rn.
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Example 6
Determine a vector in R2 that is orthogonal to (3, –1). Show that
there are many such vectors and that they all lie on a line.
Solution
Figure 4.7
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Theorem 4.3
Let u and v be vectors in Rn.
(a) Triangle Inequality:
||u + v||  ||u|| + ||v||.
(a) Pythagorean theorem :
If u.v = 0 then ||u + v||2 = ||u||2 + ||v||2.
Figure 4.8(a)
Figure 4.8(b)
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Distance between Points
Let x  ( x1 , x2 , ..., xn ) and y  ( y1 , y2 , ..., yn ) be two points in Rn.
The distance between x and y is denoted d(x, y) and is defined by
d (x, y)  ( x1 - y1 ) 2    ( xn - yn ) 2
x
Note: We can also write this distance as follows.
x-y
y
d (x, y)  x - y
Note: It is clear that d (x, y)  d (y, x)
(the symmetric property)
Example 7. Determine the distance between the points
x = (1,–2 , 3, 0) and y = (4, 0, –3, 5) in R4.
Solution
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Homework
Exercise set 1.5 pages 47 to 48:
3, 7, 8, 9, 11, 13, 16, 17, 26.
Exercise 36
Let u and v be vectors in Rn.
Prove that ||u|| = ||v|| if and only if u + v and u - v are orthogonal.
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