SOME TERMINOLOGY - MA 221
List of terms: linear combination of vectors, span, subspace,linearly dependent, linearly
independent, basis, dimension, domain and range of a matrix, column space, row space,
rank, null space.
LINEAR COMBINATION
If T is a set of vectors, say T = {v1 , v2 , . . . , vk }, then a linear combination of the vectors in
T is simply a vector of the form α1 v1 + α2 v2 + . . . + αk vk where the α’s are any real numbers.
In other words, we simply multiply the vectors by constants and add.

 

 1
 
Example: Let T = 
 3 ,


2

5 

1 
 then a couple linear combinations of vectors in T

−3 
would be












1
5
22
1
5
−7











2  3  + 4  1  =  10  and 3  3  − 2  1  =  7 

2
−3
−8
2
−3
12
Notice that if T has just one vector, then a linear combination would just be a multiple of
the vector.



1 




Example: Let T =  −3  , then a couple linear combinations would be



2 








1
3
1
−4








3  −3  =  −9  and − 4  −3  =  12 
2
6
2
−8
In the trivial case when the vector is the zero vector, then the all the linear combinations
would still be the zero vector.
SPAN
If T is a set of vector, say T = {v1 , v2 , . . . , vk }, then the span of T, written span(T )
is the set of all linear combinations of vectors in T. Or more mathematical, span(T ) =
{α1 v1 + α2 v2 + . . . + α
R} .
k v
k : αi ∈
 







1
1








Example: Let T =  −3  , then span(T ) = α  −3  : α ∈ R which is a line







2 
2
(Note that α can be zero, positive or negative, so we get the line through the origin and the
vector).

 

 







5 
1
5
 1




 





Example: If T =  3  ,  1  , then span(T ) = α  3  + β  1  : α, β ∈ R







2
−3 
2
−3
is a plane. Again since both parameters can be zero, it is the plane generated by the origin
and the two vectors. Note that it’s possible to have redundancy in the set of vectors.


 



1
2 
1





 

Example: Let T =  −3  ,  −6  , then span(T ) is the same as span −3


2
4 
2

1










since multiples of the second vector will be along the same line as generated by  −3  .
2
Similarly we can haveredundancy
when
generating
a
plane.

 
 


5
6 
 1


 
 

Example: Let T =  3  ,  1  ,  4  . Here the span is the same as the span



2
−3
−1 
of the Þrst two vectors since the third is the sum of the Þrst two. Thus the third vector is
already on the plane (i.e. the span) generated by the Þrst two vectors.
If the span is all of R2or R3 , etc,
 then
 we
 would
 say so.


1
5
2



 
 

Example: Let T =  3  ,  1  ,  1  , then span(T ) = R3 .



2 −3
13 
1
5
,
, then span(T ) = R2 .
Example: Let T =
3
1
1
0
Example: Let T =
,
, then span(T ) = R2 .
0
1
SUBSPACE
A subspace of R2 , R3 , etc is a subset in which the sum of vectors in the subset is back in the
subset and multiples of vectors in the subset is back in the subset. We say that subspaces are
closed under addition of vectors and scalar multiplication of vectors. More mathematically,
M is a subspace if (1) x and y are in M, then x + y is in M and (2) if x is in M and α is a
scalar (i.e. a number) then αx is in M. It’s easy to see that the origin must be in M (i.e. let
α be the number
0). You can also convince yourself that the subspaces of R2 are the trivial
−
→
subspace 0 , lines through the origin, and the entire space R2 . In R3 , you need to add to
the list planes through the origin. If a line doesn’t go through the origin, then we will say
that it is a translate of a subspace (i.e it is parallel to a line through the origin) - similarly
for planes. In higher dimensions, we could things like 7 dimensional subspace of R13 .
Please notice that the span of any set of vectors is a subspace.
LINEARLY INDEPENDENT/DEPENDENT
Loosely and intuitively, a set of vectors is linearly independent if no one of them is a linear
combination of the others;
the
 otherwise
 
set linearly dependent. First some examples.

5 
 1

 

Example: Let T = 
 3  ,  1  . T is a linearly independent set of vectors since



2
−3 
neither vector is a multiple
the other.

of 


2 
 1


 

Example: If T =  3  ,  6  . T is a linearly dependent set of vectors since the



2
4 
second vector is 2 times the Þrst. Or we could also say that the Þrst is one half the second.
Notice that if the zero vector is a member of the set, then the set must be linearly dependent
since the zero vector
times
is 0 
 any
of the other vectors.

0 
 1

 

Example: If T = 
 3  ,  0  . T is a linearly dependent set of vectors since the zero



2
0 
vector is 0 times the Þrst
 vector.
 
 


5
6 
 1


 
 

Example: Let T =  3  ,  1  ,  4  . T is linearly dependent since the third


2
−3
−1 
vector is the sum of the Þrst two vectors. Clearly we could also use the fact that the Þrst
vector is the difference of the other two.
BASIS
If M is a subspace and M = span(T ), then T is a basis for M if T is linearly independent
set of vectors. In other words, a basis is a set of vectors which span M and which has no
redundancy.


 
 

 
 



5
6 
5
6 
 1
 1




 
 

 
 

Example: Let M =span 3  ,  1  ,  4  . The set of vectors  3  ,  1  ,  4 


2
−3
−1 
2
−3
−1 
is not a basis since it is linearly dependent. However, if we throw away one of the vectors,
say the third one, then the resulting set is a basis for M. A basis is not unique. In this case
the Þrst two vectors, the Þrst and third, or the second and third would all be bases for the
M. We can say that a basis generates M and furthermore all the vectors in the basis are
needed.

 
 


1
0
0 


1
0

 
 

,
and for R3 it’s  0  ,  1  ,  0  .
The standard basis for R2 is
0
1

0
0
1 
You can see the pattern for higher dimensional spaces.
DIMENSION
The dimension of a subspace M is the number of vectors in a basis for M. Indeed, there is
something to prove here; namely, that any two bases for a subspace have the same number
of vectors. When you take linear algebra, you will see such a proof.
Our intuition tells us that the dimension of a line is 1, and so does the mathematics. That
→
→
is, given a line through the origin, we can represent it as {α−
v } for any non-zero vector −
v
−
→
on the line. So { v } would be a basis for the line and since there is one vector in the basis,
the dimension is one. A similar argument works for a plane.
DOMAIN and RANGE of a MATRIX
Just as in high school, if you think of a matrix as a function (i.e. as a transformation), then
the matrix should have a domain and range. For example if A is a 3 × 2 matrix, then as
discussed in class we write A : R2 −→ R3 which means that if v ∈ R2 , then A sends v to
Av which is in R3 . So the domain of A would be all of R2 . That is, there are no restrictions
on which vectors A can multiply except that they have two components. But what about
the range (recall from high school that the range will be all the “y” values). First of all the
range will be a subset of R3 . It need not be all of R3 . Furthermore, the range is a subspace
(i.e. Av + Au = A(v + u) and αA(v)
= A(αv)). So the range must be one of the subspaces of
−
→
3
R , namely, the trivial subspace 0 , a line through the origin, a plane through the origin,
or the entire space R3 . The answer will depend on the entries of A. But to understand this
a bit better, notice how the range is related to the columns. Let


x
y
be any vector in R2
a d


and let A =  b e  Then
c f
A
x
y





 





a d ax + dy
ax
dy
a
d

 x



 





= b e 
=  bx + ey  =  bx  +  ey  = x  b  + y  e 
y
c f
cx + f y
cx
fy
c
f
which says that the range of A is the span of the columns of A. Mathematically, we could
write
R(A) = span({columns of A})
where we use R(A) for
the range
of A and D(A) for the domain of A.


1
5

1 
Example: Let A =  3
. Since A is 3 × 2, D(A) = R2 and R(A) ⊂ R3 . But by
2 −3
the above argument, the range is the span of the columns of A and since the columns are
linearly independent,the range
 of A is a plane.
1 2


Example: Let A =  3 6  . In this case, the range of A is a line in R3 .
2 4


1
5 3

1 1 
Example: Let A =  3
. In this case, the three columns are linearly independent.
2 −3 1
Thus the range is all of R3 .
3
The only way to get 
the trivial
 subspace of R as a range is to have the zero matrix.
0 0


Example: Let A =  0 0  . Here the domain is R2 and the range is the origin in R3 .
0 0
COLUMN SPACE of a MATRIX
The column space of a matrix is span of the columns. So think of each column in a matrix
as a vector and take the span of the set of these vectors. This will be the column space of
the matrix. With the work above we have for a matrix A
Column space of A = R(A) = span {columns of A}


1
5

1 
Example: Let A =  3
 , then
2 −3

 

 1

 
Column space of A = R(A) = span  3  , 


2
which is a plane in R3 .

5 

1 


−3 


1 2


Example: Let A =  3 6  , then
2 4

 


2 
 1


 

Column space of A = R(A) = span  3  ,  6 



2
4



 1 



which is a line in R3 . Furthermore a basis for the column space is  3  .



2 
Example: Let A =
1
2
2 −3
, then
1
2
Column space of A = R(A) = span
2
−3
,
which is all of R2 . So a basis for the column space would be
1
0
,
0
1
.
ROW SPACE of a MATRIX
The row space of a matrix is span of the rows. So think of each row in a matrix as a vector
and take the span ofthe set ofthese vectors. This will be the column space of the matrix.
1
5
1 
Example: Let A = 
 3
 , then
2 −3
Row space of A = R(A ) = span
1
5
,
3
1
,
2
−3
which will be all of R2 .
Recall that the transpose of a matrix (written A ) is the matrix obtained from A by interchanging the columns and rows of A. Thus the row space of A will be the column space of
A .
RANK of a MATRIX
The rank of a matrix is the dimension of the range of the matrix. Or since the range of
the matrix is the same as its column space, we could say that the rank of a matrix is the
dimension of its column space. Furthermore, it is true (even though we will not give a proof)
that the dimension of the column space of a matrix is always equal to the dimension of the
row space, we could even say that the rank of a matrix is the dimension of its row space.
Somtimes you will see the terms column rank and row rank in textbooks. Since the dimension
of the column space is the same as the dimension of the row space, these two terms (column
rank and row rank) are the same.
Summarizing,
Rank(A) = dim(R(A)) = dim(column space of A) = dim(row space of A) = dim(R(A ))
NULL SPACE
If A is an m × n matrix, the null space of A (written N (A)) is the subspace of all vectors in
→
Rn such that A−
x = 0. In other words, the null space consists of all vectors which A sends
to the zero vector in Rm . 1 2
−2
, then N (A) = α
. To Þnd the null space, you will
Example: Let A =
2 4
1
−
→
→
need to solve the equation A−
x = 0 , which can be done using Gaussian elimination.
One of the major theorems in linear algebra (often called the fundamental theorem of linear
algebra) is
If A is m × n, then dim(N (A)) + dim(R(A)) = n
Another important theorem is
If A is m × n, then N (A)⊥R(A ) and N (A )⊥R(A)
Rather than giving proofs of these theorems, we will illustrate them in class with examples.