1.3 Vector Equations
Vectors in R2
: 1-column matrices or
ordered pairs of real numbers
Example: Given
1
u   
2
and
3
v    ,
1 
3u,  2v, and 3u  (2)v.


u
v
Geometric Interpretation?


find
Vectors in R3
Vectors in Rn
: 31 column matrices
or points in three-dimensional space
:
n 1 column matrices
 a1 

  

 a2 

n
R  u 
: ai  R
  M

 


 an 

 Two vectors are equal if and only if their corresponding
entries are equal.
 The vector cu(where c is a real number) is a scalar
multiple of


u.
Algebraic Properties of R n
n
For all u, v, w in R and all scalars c and d :
i) u  v  v  u
v) c(u  v)  cu  cv

ii) ( u  v)  w  u  (v  w)
vi) ( c  d)u  cu  du
iii) u  0  0  u  u
vii) c(du)  (cd)u
iv) u  (-u)  -u  u  0
viii) 1  u  u
(Note that -u  (-1)u)
Linear Combinations
v1,v2 ,L ,v p Rn
c1,c 2 ,L ,c p  R,
Given vectors
and
given scalars
c1v1  c 2 v 2 L  c p v p
linear combination of
is a
v1,v 2 ,L ,v p
with weights
c1 , c2 ,  , c p .

Example:
1 
2v1  v 2
2
v1
0

Example: Determine whether w can be generated as a linear
Combination of v 1 and v2 , where
1
1
v1   , v 2   ,
2
1 


and
6 
w   .
3
A vector equation
x1v1  x 2 v 2 L  x p v p  w
has the same solution set as the linear system whose
augmented matrix is

v
1

v2 L
vp | w
w can be generated by a linear combination of vectors in
v ,v ,L v 

1
2
p
if and only if the following linear system is consistent:

v
1
v2 L

vp | w
Definition
n
v
,v
,L
v

R
If
, then the set of all linear
1
2
p
combinations of v1, v 2 ,L v p is denoted by


Span v1,v2,L v p and is called the
subset
 of
R n spanned by
v ,v ,L v 
1
2
p





Span{v1,v2,L v p }  c1v1  c 2v2 L  c p v p : c i R
Geometric Description of Span:
 1  
Span   
  2 
1   1 
Span  ,   
  2  1  
1   1 
    
Span 2, 1 
 
 0   0  
    