1.3 Vector Equations
New definition fo vector: A vector is a
matrix with only one column.
Ex:
 1
 2  in R2.
 
In general, Rn :
 v1 
v 
 2
v  v3 
 

vn 
A vector still has magnitude and
direction, but here it is more convenient
to think only of the endpoints.
The zero vector 0 has as many 0’s as
dimensions.
0 
0   
0     0 
0

 0 
Example:
 
Addition
Add vectors entry by entry.
1
u 
Example:
3
 2
v 
1
1  2 3
u v  
 

3  1 4
Parallelogram Rule: If u and v are in R2
are represented as points in the plane,
then u + v corresponds to the 4th vertex
of the parallelogram whose other
vertices are 0, u and v.
Scalar Multiplication: a scalar is a
constant number.
 cu1 
 u1 
cu   
u 
Let
cu2 
u2  , then
1 
 2
u 
2u   
Ex: Let
2 , then
 4
3

3
 

 u 2
2
  3
 
.
Linear Combinations
Once we can combine vectors and
multiply them but scalars, we can talk
about linear combinations of them.
v
,
v
,

,
v
1
2
p in a
Def: Given vectors
row is in Rn, and scalars c1,c2 ,, c p , the
vector y defined by
y  c1v1  c2 v 2    c p v p is called a
linear combination of v1 , v 2 ,, v p with
weights c1,c2 ,, c p . Note: p ≤ n.
Examples of linear combinations of
2
v1 , v 2 : 2v1  3v 2 , 3 v1  3v 2 , v 2 , 0.
Geometrically: write the following as
linear combinations of
 2
 2

v1   
v2   
1 and
2
0
6

4


a  b  c 
 3 ,
6 , and
 1 ,
7
d 
 4
a  v1  v 2 , b  v1  v 2
c  4v1  v 2 , d  v1  2.5v 2
Back to algebra!
In R3: Let
3
1
4






a1  0 a2  2 a3   6 
 
10
3
14
  1
b   8 
 5
Is b a linear combination of the ai?
OR: are there scalars
x1 , x2 , x3
that x1a1  x2a2  x3a3  b ?
such
Write the vector equation:
1 
4
 3    1
x1 0  x2  2   x3  6    8 
 
   
3
14
10  5
Distribute the xi to get a system of
equations.
Use rref to solve the system to get:
(1, –2, 2)
Yes, b is a linear combination of the ai.
Review: The vectors a1 , a2 , a3 , b are
columns of an augmented matrix:
a1 a2 a3 b . The solution to
x1a1  x2a2  x3a3  b is found by solving
the linear system whose augmented
matrix a1 a2 a3 b .
In general, b can be generated by a
linear combination of a1 , a2 ,..., an iff  a
solution to the linear system
corresponding to the augmented matrix
a1 a2  an b .
Note:  infinitely many vectors that can
be generated by the ai since  infinitely
many values of the xi.
Def: suppose v1 , v 2 ,..., v p are in Rn;


then Span v1 , v 2 ,..., v p = the set of all
linear combo’s of v1 , v 2 ,..., v p .


Or Span v1 , v 2 ,..., v p is the collection
of all vectors that can be written as
x1v1  x2 v 2    x p v p where the
xi are scalars.
Example: in R3, Let
 3


v   4
5
Span{v} is just the set of all scalar
multiples of v, the line through v and 0.
Note: Span{u, v} is a plane in through u,
v, and 0 in R3 in the illustration of Rn in
general.