Chapter 1
Matrices
Definition 1.1. An nk matrix over a field F is a function A:{1,2,…,n}{1,2,…k}F.
A matrix is usually represented by (and identified with) an nk (read as “n by k”)
array of elements of the field (usually numbers). The horizontal lines of a matrix are referred
to as rows and the vertical ones as columns. The individual elements are called entries of the
matrix. Thus an nk matrix has n rows, k columns and nk entries. Matrices will be denoted by
capital letters and their entries by the corresponding small letters. Thus, in case of a matrix A
we will write A(i,j)=ai,j and will refer to ai,j as the element of the i-th row and j-th column of
A. On the other hand we will use the symbol [ai,j] to denote the matrix A with entries ai,j.
Rows and columns of a matrix can (and will) be considered vectors from Fk and Fn,
respectively, and will be denoted by r1,r2, … rn and c1,c2, … ,ck. The expression nk is called
the size of a matrix.
Definition 1.2. Let A be an nk matrix. Then the transpose of A is the kn matrix AT, such
that for each i and j, AT(i,j)=A(j,i).
In other words, AT is what we get when we replace rows of A with columns and vice
versa..
Proposition 1.1. For any n,kN the set of MTR(n,k) all nk matrices over F with ordinary
function addition and multiplication of functions by constants is a vector space over F,
dim(MTR(n,k))=nk.
Proof. MTR(n,k) is a vector space (see Example 4.4???). As a basis for MTR(n,k) we can use
1 if (i, j )  ( p, q)
the set consisting of matrices Ap,q, where A p.q (i, j )  
0 otherwise
The operation of matrix multiplication is something quite different from the
abovementioned operations of matrix scaling and addition. It is not inherited from function
theory, it is a specifically matrix operation. The reason for this will soon become clear.
Definition 1.3. Suppose A is an nk matrix and B is a ks matrix. Then the product of A by B
k
is the matrix C, where ci,j=ai,1b1,j+ai,2b2,j+ … ai,kbk,j=  a i ,t bt , j
t 1
In other words, ci,j is the sum of the products of consecutive elements of the i-th row of
A by the corresponding elements of the j-th column of B. Strictly speaking multiplication of
matrices in general is not an operation on matrices as it maps MTR(n,k)MTR(k,s) into
MTR(n,s). It is an operation though, when n=k=s. Matrices with the same number of rows and
column are called square matrices.
Notice that C=AB is an ns matrix. Notice also that for the definition to work, the
number of columns in A (the first factor) must equal the number of rows in B (the second
factor). In other cases the product is not defined. This suggest that matrix multiplication need
not be commutative.
0 2
2
1  1 2 

Example 1.1. Let A  
and B   1 3 1 . then

2 0  3
 1  2 2
 2  1  2 0  3  4 2  1  4  5  7 5 
AB  
 = 
 and BA is not defined since the
4  0  3 0  0  6 4  0  6 1 6  2
number of columns of B is not equal to the number of rows of A
1 0
0 0 
0 0 
0 0
Example 1.2. Let A  
and B  
. Then AB  
while BA  



 . This
1 0
0 0 
1 1
 2 0
example proves that AB may differ from BA even when both products exist and have the
same size.
1 if s  t
Example 1.3. The nn matrix I defined as I ( s, t )  
is the identity element for
0 if s  t
n
matrix multiplication on MTR(n,n). Indeed, for every matrix A, AI(i,j) =
 A(i, t ) I (t , j ) =
t 1
A(i,j) because A(i, t ) I (t , j ) is equal to A(i,j) when t=j and is equal to 0 otherwise. Hence
AI=A. A similar argument proves that IA=A. All entries of the identity matrix I are equal to 0
except for the diagonal entries that are all equal to 1. The term diagonal entries refers to the
diagonal of a square matrix, i.e. the line connecting the upper-left with the lower-right
position in the matrix. The line consists of all elements of the form C(i,i).
Example 1.4. Suppose A is an nk matrix over F and X=[x1,x2, … ,xk] is a single-row matrix
representing a vector from Fk. By the definition of the matrix multiplication, AXT is a singlecolumn matrix representing a vector from Fn. Hence if we fix A and let X vary over Fk we
have a function A: Fk  Fn defined as A(X)=AXT.
Theorem 1.1. For every nk matrix A the function A(X)=AXT is a linear mapping of Fk into
Fn.
Proof. The i-th component of the vector A(pX) is ai,1px1+ai,2px2+ … +ai,kpxk =
p(ai,1x1+ai,2x2+ … +ai,kxk). Since ai,1x1+ai,2x2+ … +ai,kxk is the i-th component of A(X), we
have proved that A preserves scaling. Suppose Y=[y1,y2, … ,yk] and consider A(X+Y) =
A((X+Y)T) = A[x1+y1, x2+y2, … , xk+yk]T. By the definition of the matrix product, the i-th
component of A(X+Y) is equal to ai,1(x1+y1)+ai,2(x2+y2) … +ai,k(xk+yk) = ai,1x1+ai,2x2+ …
+ai,kxk + ai,1y1+ai,2y2+ … +ai,kyk and this is the i-th component of A(X)+ A(Y). This shows
that A preserves addition.
In case of coordinate vectors we will often leave out the transposition symbol T
allowing the context to decide whether the sequence should be written horizontally or
vertically.
Corollary. For every nk matrix A and for every two ks matrices B and C,
A(B+C)=AB+AC, i.e. matrix multiplication is distributive with respect to matrix addition.
Proof. From Definition 1.3 (of matrix multiplication) the i-th column of A(B+C) is the
product of A and the i-th column of B+C. Since the i-th column of B+C is the sum of the i-th
columns of B and C, by Theorem 1.1 we have that the i-th column of A(B+C) is equal to the
i-th column of AB+AC.
Theorem 1.1 says that with every nk matrix we can associate a linear mapping from
Fk into Fn. The following definition assigns a matrix to a linear mapping.
Definition 1.4. Let us consider a linear mapping :FkFn and let us choose a basis S={v1,v2,
… ,vk} for Fk and another, R={w1,w2, … ,wn} for Fn. For each vector viS there exist unique
scalars a1,i, a2,i, … , an,i such that vi=a1,i,w1+a2,i,w2 + … + an,iwn. The matrix [ai,j] is called the
matrix of  with respect to bases R and S, and is denoted by M SR ( ) .
In other words, for each i, the i-th column of the matrix M SR ( ) is equal to [(vi)]R.
Notice that the matrix assigned to a linear mapping depends on the choice of the two
bases R and S. Choosing another pair of bases we will obtain another matrix for the same
linear mapping. Also, two different linear mappings may have the same matrix but with
respect to two pairs of bases. Two matrices of the same linear mappings are called similar.
We will study similarity of matrices in the next chapter???
Example 1.5. Let :R3R2, (x,y,z) = (x+2y2z,3xy+2z). We will construct the matrix for
 in bases S={(0,1,1),(1,0,1),(1,1,0)} and R={(1,1),(1,0)}. We calculate the values of  on
vectors from S and represent them as linear combinations of vectors from R. (0,1,1) = (0,1)
=1(1,1)+(1)(1,0), (1,0,1) = (1,5) = 5(1,1)+(6)(1,0) and (1,1,0) = (3,2) = 2(1,1)+1(1,0).
Finally we form the matrix placing the coefficients of the linear combinations in consecutive
5 2
1
columns. M RS ( )  

 1  6 1 
Recall that [v]R denotes the coordinate vector of v with respect to a basis R. The
following theorem, states that matrices and linear mappings are essentially the same thing, or
rather, there is a 1-1 correspondence between matrices and linear mappings.
Theorem 1.2. Given a linear mapping :FkFn and bases S={v1,v2, … ,vk} for Fk, and
R={w1,w2, … ,wn} for Fn, for every vector x from Fk and for every kn matrix A we have
A[ x]S  [ ( x)] R if and only if A  M RS ( )
Proof. () Suppose that A  M RS ( ) =[ai,j] and [x]S=[x1,x2, … ,xk]. We have (x) =
k
 ( x s v s ) =
s 1
k
 x s (v s ) =
s 1
k
n
s 1
t 1
 xs ( at , s wt ) =
k
n
 xs at ,s wt =
s 1 t 1
n
k
t 1
s 1
 ( xs at , s )wt (the proof of
the last equality is left to the reader) which means that the t-th coordinate of [(x)]R =
k
 xs at ,s . On the other hand, the t-th coordinate of M RS ( )[ x]S is also equal to
s 1
k
a
r 1
x .
t ,r r
() Now suppose that for every x from Fk, A[ x]S  [ ( x)] R . Let us put x=vi. Then [vi]S =[0,
… ,1, … ,0]. The only 1 in the sequence is on the i-th place. Hence A[vi]S is equal to the i-th
column of A. By the remark following Definition 1.4 we are done.
Theorem 1.3. If :FkFn and :FnFp are linear mappings, and S,R,T are bases for
Fk,Fn,Fp, respectively, then M TS (   )  M TR ( ) M RS ( ) .
Proof. This is an immediate consequence of Theorem 1.2.
Theorem 1.4. Matrix multiplication is associative, i.e. for any three matrices A,B and C
A(BC) = (AB)C – provided the products exist.
Proof. This follows immediately from Theorem 1.3 and the fact that function composition is
associative.
Definition 1.5. Let A  M RS ( ) . Then the rank of A is the rank of .