MATRICES
ROHAN RAMCHAND
We will start by defining the concept of a matrix.
Definition 1. Let F be a field and let m, n ∈ N≥0 . Then an m × n F -matrix is a table
consisting of m rows and n columns with elements in F .
We defined in previous sections (see Linear Transformations) the matrix space as
(Matm×n (F ), +, α),
or the set of all m × n F -matrices, along with matrix addition and scalar multiplication.
Furthermore, dim Matm×n (F ) = mn. The proof of this statement follows.
Proof. We begin by defining the standard basis of Matm×n (F ) as the set of matrices
E = {E0,0 ..., Em,n }, where Em,n = (ei,j ) 1≤i≤m such that ei,j = δm,n (i, j). Then A =
1≤j≤n
P
P
i,j αi,j Ei,j , where αi,j = Ai,j ; therefore, E spans A. Furthermore, if
i,j αi,j Ei,j = 0,
then αi,j = 0 and E is linearly independent. Therefore, E is a basis with length mn; then
dim Matm×n = mn and the proof is complete.
We now define one final operation on matrices: matrix multiplication.
Definition 2. Let A ∈ Matm×r (F ) and B ∈ Matr×n (F ). Then the product of A and B,
C = A · B ∈ Matm×n (F ), is defined as
ci,j =
r
X
ai,k bk,j .
k=1
Matrices and Linear Transformations
The following two properties of matrices are used in the proofs that follow and are stated
without proof.
Distributivity: Let A ∈ Matm×r (F ) and B1 , B2 ∈ Matr×n (F ). Then
A · (B1 + B2 ) = A · B1 + A · B2 .
Scalar Multiplication: Let A ∈ Matm×r (F ), B ∈ Matr×n (F ), and λ ∈ F . Then
A · (λ · B) = λ · (A · B).
1
MATRICES
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We now define the following map.
Definition 3. Let A ∈ Matm×n (F ). Define
TA : Matn×r → Matm×r : TA (B) = A · B.
Theorem 1. TA is a linear transformation.
Proof. Both properties of matrix multiplication above satisfy the properties of a linear
transformation.
We will, in particular, study the case r = 1, defined below.
Definition 4. Let A ∈ Matm×r and let r = 1. Then A is a column vector. In particular, the set of m-dimensional column vectors is referred to as colm (F ). Furthermore,
colm (F ) can be identified with F m , the former being a column vector and the latter being
(canonically) a row vector.
Therefore, we redefine TA as TA : Matm×1 → Matn×1 : F n → F m .
We will now revisit the concept of matrix multiplication as composition of linear transformations. Let
a11 . . . a1n
..
..
A = ...
.
.
am1 . . . amn
and let
x1
~x = ... .
xn
Then
a11 . . . a1n
x1
.. ..
..
TA (~x) = ...
.
. .
am1 . . . amn
an
a11 x1 + a12 x2 + ... + a1n xn
a21 x1 + a22 x2 + ... + a2n xn
=
..
.
am1 x1 + am2 x2 + ... + amn xn
a12
a1n
a11
a2n
a22
a21
= x1 .. + x2 .. + ... + xn ..
.
.
.
am1
am2
amn
MATRICES
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This is one of the more important results of this course: the product of a matrix and
a vector is a linear combination of the columns of the matrix, where the coefficients in
the combination are the elements of the vector. This implies that the result of matrix
multiplication is a linear transformation in and of itself, and therefore TA , as defined
above, sends Matm×n (F ) to Hom(F n , F m ).
We now state an important theorem regarding TA .
Theorem 2. TA is a bijective linear transformation; in other words, Matm×n (F ) and
Hom(F n , F m ) are isomorphic.
Proof. Let A, A1 , A2 ∈ Matm×n (F ) and λ ∈ F .
TA1 +A2 (~x) = (A1 + A2 )~x
= A1 ~x + A2 ~x
= TA1 (~x) + TA2 (~x)
TλA (~x) = (λA)(~x)
= λ(A~x)
= λTA (~x)
Therefore, TA is a linear transformation.
Let M = T −1 : Hom(F n , F m ) → Matm×n (F ). For a linear transformation A : F n → F m ,
MA is a matrix, composed of columns m
~ 1 , ..., m
~ n , such that
MA · ~x = A(~x) =
n
X
xi m
~ i.
i=1
By definition, the basis (e1 , e2 , ..., en ), with ei = (δi (1), δi (2), ..., δi (n)) spans any vector
space V : therefore,
A(~x) = A · (x1 e1 + ... + xn en ) =
n
X
xi A(ei ).
i=1
Therefore,
MA = (A(e1 ), ..., A(en ))
and MA is defined for all A; therefore, TA has an inverse for all linear transformations A
and is therefore an isomorphism.
Note that we did not actually prove that M is a linear transformation; the proof of this
statement is left as an exercise.
This concept is illustrated with the following example.
MATRICES
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Example. Let
A : R2 → R2 : A(x, y) =
Then
x+y
x−y
MA = (A(1, 0), A(0, 1)) =
1 1
1 −1
.
.
It has been stated on multiple occasions that composition of linear transformations is equivalent to matrix multiplications; this theorem is restated formally and proven below.
Theorem 3. Let A ∈ Matm×r (F ), B ∈ Matr×n (F ), with AB ∈ Matm×n (F ). Then let
TA : F r → F m , TB : F n → F r , with TAB : F n → F m . Then
TAB = TA ◦ TB .
Proof. As a convenience measure, we will denote the ith column of a matrix M as Mi .
TAB (~x) = (A · B) · ~x
=
n
X
xi (A · B)i
i=1
=
n
X
xi (A · Bi )
((A · B)i = A · Bi )
i=1
=
n
X
A(xi Bi )
i=1
=A·
n
X
xi Bi
i=1
= A · (B · ~x)
= (TA ◦ TB )(~x)
Armed with this theorem, we can prove another theorem of matrix multiplication.
Theorem 4. Let A ∈ Matm×r (F ), B ∈ Matr×l (F ), C ∈ Mat l × n(F ) be matrices. Then
(A · B) · C = A · (B · C).
Proof.
(1)
T(A·B)·C = TA·(B·C)
(2)
TA·B ◦ TC = TA ◦ TB·C
(3)
(TA ◦ TB ) ◦ TC = TA ◦ (TB ◦ TC )
By associativity of composition of functions, (3) is true and the proof is complete.
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