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Linear Algebra
Chapter 4
Vector Spaces
4.1 The vector Space Rn
Definition 1:
…………………………………………………………………….
The elements in Rn called ………….
Addition and scalar multiplication:
Definition 2:
Let u  ....................... and v  ....................... be two elements of Rn.
k scalar.
1) u  v  ........................................
2) u  v  ...........................................
3) k u  ..............................................
Addition
scalar multiplication
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4.1 The vector Space Rn
Ex 1:
Let u   1, 4,3,7  and v   2, 3,1,0 be vectors in R 4 .
Fined:
u v 
3u 
Note:
*) 0   0, 0,..., 0  having n zero called ............................. in R n .
*) The vector  (1)u 
called ................... of u .
Ex : u   2,3, 1  R 3   u  .....................
*) if u ,v R n then u v  .............
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Theorem 4.1
Properties of vectors Addition and scalar multiplication:
Let u, v, and w be vectors in Rn and let c and d be scalars.
1) u + v =
5) c(u + v) =
2) u + (v + w) =
6) (c + d) u =
3) u + 0 =
7) c (d u) =
4) u + (–u) =
8) 1u =
Ex2:
Let u = (2, 5, –3), v = ( –4, 1, 9), w = (4, 0, 2) in R3.
Determine the linear combination 2u – 3v + w.
Solution
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Column Vectors
Row vector: u  (u1 , u2 , ..., un )
Column vector:
u1 
u   
u n 
Then:
u1  v 1 
      ..........
   
u n  v n 
and
 u1 
c    ..........
u n 
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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 ………………. of u and v is denoted …….. and is defined
by:
Example 3
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 =
2. (u + v).w =
3. cu.v =
4. u.u …… , and u.u = …….  u = …..
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Norm of a Vector in Rn
Definition
The norm of a vector u = (u1, …, un) in Rn is:
……………………………………..
Definition
A unit vector is a vector whose norm is …... (………)
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 …….………..…….
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Example 4
Find the norm of each of the vectors u = (2, -1, 3) of R3
and v = (3, 0, 1, 4) of R4.
Normalize the vector u.
Solution
……………………………………………………………………..
……………………………………………………………………..
……………………………………………………………………..
……………………………………………………………………..
Example 5
Show that the vector u=(1, 0) is a unit vector in R2.
Solution
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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
cos   ...............
, ........    ......
Example 6
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 …………….. if the angle between them
is a right angle .
Theorem 4.2
Two nonzero vectors u and v are orthogonal 
Example 7
Show that the vectors u=(2, –3, 1) and v=(1, 2, 4) are orthogonal.
Solution
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Note
(1, 0), (0,1) are orthogonal unit vectors in R2.
………., ………., ………. are orthogonal unit vectors in R3.
………., ………., … , ……….,are orthogonal unit vectors in Rn.
Distance between Points
Let x  ( x1 , x2 , ..., xn ) and y  ( y1 , y2 , ..., yn ) be two points in Rn.
The …………… between x and y is denoted ….... and is defined by:
Note: It is clear that d (x, y)  ..............
Example 8. Determine the distance between
x = (1,–2 , 3, 0) and y = (4, 0, –3, 5) in R4.
Solution
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Properties of Norm:
Properties of Distance:
1) u
1) d  x , y 
2) u  0 
2) d  x , y   0 
3) k u 
3) d  x , y  
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