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Composition of linear transformations
and matrix multiplication
Math 130 Linear Algebra
D Joyce, Fall 2015 Throughout this discussion, F refers to a fixed field. In application, F will usually be R. V , W , and X will be vector spaces over F.
Consider two linear transformations V T → W and W S → X where the codomain of one is the same as the domain of the other. Their composition V −→S◦T X is illustrated by the commutative diagram
V W
X
T -
?
S
@ @ @ @ @R
S ◦ T
As each of T and S preserve linear combinations, so will the composition, so S ◦ T is also a linear transformation.
Coordinates again. When the vector spaces are coordinatized, that is, when we have chosen a ba- sis β for V , γ for W , and δ for X, we have iso- morphisms φβ : V →' F p, φγ W →' F n, and
φδ : X ' → F m. Although we could do everything explicitly with these isomorphisms, they really get in the way of understanding. So instead, let’s just assume that the vector spaces actually are F p, F n, and F m, and we have two linear transformations T : F p^ → F n and S : F n^ → F m. Then F p^ T → F n^ is represented by an n × p matrix
B, F n^ →S F m^ is represented by a m × n matrix A,
and their composition F p^ S◦T −→ F m^ is represented by some m × p matrix. We’ll define matrix multipli- cation so that the product of the two matrices AB represents the composition S ◦ T.
F p^ F n
F m
B -
?
A
@ @ @ @ @R
AB
Let’s see what the entries in the matrix product AB have to be. Let v be a vector in F p, then w = T (v) is a vector in F n, and x = S(w) = (S ◦ T )(v) is a vector in F m. The n × p matrix B represents T. Its jkth^ entry is Bjk, and it was defined so that for each j,
wj =
k
Bjkvk.
Likewise, the m × n matrix A represents S. Its ijth^ entry is Aij , and it was defined so that for each i, xi =
j
Aij wj.
Therefore
xi =
j
Aij
k
Bjkvk =
k
j
Aij Bjk
vk.
Definition 1. Given an m × n matrix A and an n × p matrix B, we define AB to be an m × p matrix whose ikth^ entry is
(AB)ik =
j
Aij Bjk.
With this definition, matrix multiplication corre- sponds to composition of linear transformations.
A mnemonic for multiplying matrices. Al- though the equation (AB)ik =
j Aij^ Bjk^ is fine for theoretical work, in practice you need a better way to remember how to multiply matrices. The entry Aij in a row of the first matrix needs to be multiplied by the corresponding Bjk in a column of the second matrix. If you place the matrix A to the left of the product and place the matrix B above
the product, it’s easier to see what to multiply by what. Take, for instance, the following two 3 by 3 ma- trices.
A =
, B =
Think of A as being made of three row vectors and B as being made of three column vectors.
A =
, B =
To get an entry for the product, work with the row in A to the left of it and the column of B above it. For example, the upper left entry of the product, work with the first row of A and the first column of B; you’ll get 4 · 2 + 5 · 0 + 6 · (−2) = −4.
Systems of linear equations are linear matrix equations. We’ll have a lot of uses for matrix multiplication as the course progresses, and one of the most important is the interpretation of a system of linear equations as a single matrix equation. Take, for example, the system of equations
5 x + 2y = 12 3 x − y = 5 x + 3y = 5
Let A be the coefficient matrix for this system, so that
A =
and let b be the constant matrix (a column vector) for this system, so that
b =
Finally, let x be the variable matrix for this system, that is, a matrix (another column vector) with the variables as its entries, so that
x =
[
x y
]
Then the original system of equations is described by the matrix multiplication Ax = b:
[
x y
]
In general, each system of linear equations corre- sponds to a single matrix equation
Ax = b
where A is the matrix of coefficients in the sys- tem of equations, x is a vector of the variables in the equations, and b is a vector of the constants in the equations. This interpretation allows us to interpret something rather complicated, namely a whole system of equations, as a single equation.
Matrix products in Matlab. If A and B are two matrices of the right size, that is, A has the same number of columns that B has rows, then the expression AB gives their product. You can com- pute powers of square matrices as well. If A is a square matrix, then A^3 computes the same thing as AA*A.
Categories. Categories are higher order alge- braic structures. We’ll look at a couple of cate- gories. One will be the category of vector spaces and linear transformations over a field, the other the category of matrices over a field F. We’ll also
Isomorphisms in the category of sets are bijec- tions.
Example 4 (The category of vector spaces VF ). Fix a field F. The objects in the category VF are vector spaces over a F and the morphisms are linear transformations. Different fields have different cat- egories of vector spaces. Hom(V, W ) is the vector space of linear transformations V → W. Since it’s a vector space over F itself, it’s actually an object in the category. Isomorphisms in the category of vector spaces are what we’ve been calling isomorphisms.
Example 5 (The category of matrices MF ). We’d like the matrices over a fixed field F to be the mor- phisms in this category. Composition will then be multiplication of matrices. But then, what are the objects? The objects in MF are the vector spaces F n^ for n = 0, 1 , 2 ,.. .. A morphism F n^ → F m^ is an m × n
matrix A. The composition of two matrices F p^ B →
F n^ and F n^ →A F m^ is the matrix product F p^ −→AB F m as we defined it above. The identity morphism F n^ → F n^ is the n × n identity matrix I with 1’s down the diagonal and 0’s elsewhere. Hom(F n, F m) is the set of matrices we’ve de- noted by Mmn.
The category MF of matrices is can be inter- preted as a subcategory of the category of vector spaces VF. It doesn’t include all the vector spaces, as infinite dimensional vector spaces aren’t objects of MF. Furthermore, MF doesn’t have any finite dimensional vector spaces except those of the form F n. We know, however, that every vector space V of finite dimension n is isomorphic F n. Note that the only isomorphisms F n^ → F m^ in MF occur when n = m.
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