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1.5 Multiplication of Vectors by Matrices
In the previous section we saw that a real valued linear function z = f(x) of a column vector x = had the form f(x) = ax where a = (a1, a2, …, an) is a row vector. In this section we extend multiplication to a matrix times a vector. This operation gives a linear function where the values of the function are column vectors.
Definition 1. Suppose A is an m(n matrix, x = and p = (p1, p2, …, pm).
a. Ax is the column vector with m components obtained by multiplying each row of A by x, i.e. the ith component of Ax is the ith row of A times x.
(Ax)i = (Ai,●)x =
b. pA is the row vector with n components obtained by multiplying p by each column of A, i.e. the jth component of pA is p times the jth column of A.
(pA)j = p(A●,j) =
Note that
# columns of A = # components of x
# rows of A = # components of p
Example 1.
Ax = = , (2, 3) , (3, 5) )) = =
For each row of A you go across that row of A and down x multiplying corresponding components and adding to get the corresponding component of Ax.
pA = (2, 3, 5) = ((2, 3, 5) , (2, 3, 5) )
= (10 + 6 + 15, 14 + 9 + 25) = (31, 48)
As with the product of a row vector and a column vector, the product of a matrix time a vector is useful for describing linear functions.
Example 2. The three linear functions
( = 2x + y + z
( = 4x – 6y
( = - 2x + 7y + 2z
can be written either as
= or w = Au
or
((, (, () = (x, y, z) or q = pB
where u = , w = , p = (x, y, z), q = ((, (, ()
A = B =
Similarly, the three linear equations
2x + y + z = 5
4x – 6y = - 2
- 2x + 7y + 2z = 9
can be written either as
= or Au = b
or
(x, y, z) = (5, - 2, 9) or pB = c
where b = and c = (5, - 2, 9).
Example 3. In Example 4 in section 1.4 an electronics company made two types of circuit boards for computers, namely ethernet cards and sound cards. Each of these boards requires a certain number of resistors, capacitors and transistors as follows
| |ethernet card |sound card |
|resistors |5 |7 |
|capacitors |2 |3 |
|transistors |3 |5 |
Let
e = # of ethernet cards the company makes in a certain day
s = # of sound card the company makes in a certain day
r = # of resistors needed to produce the e ethernet cards and s sound cards
c = # of capacitors needed to produce the e ethernet cards and s sound cards
t = # of transistors needed to produce the e ethernet cards and s sound cards
pr = price of a resistor
pc = price of a capacitor
pr = price of a transistor
pe = cost of all the resistors, inductors and transistors in an ethernet card
ps = cost of all the resistors, inductors and transistors in an sound card
Then we had the following linear functions.
r = 5e + 7s = (5, 7) = (5, 7) x
c = 2e + 3s = (2, 3) = (2, 3) x
t = 3e + 5s = (3, 5) = (3, 5) x
pe = 5pr + 2pe + 3pt = (pr, pc, pt) = p
ps = 7pr + 3pe + 2pt = (pr, pc, pt) = p
where
x =
p = (pr, pc, pt)
If we group r, c and t into a column vector y then we have
y = = = , (2, 3) , (3, 5) )) = x = Ax
where
A =
The point is that a set of linear equations can be represented compactly by a single vector matrix equation y = Ax. Similarly, if we group pe and ps into a vector q then one has
q = (pe, ps) = (5pr + 2pe + 3pt, 7pr + 3pe + 2pt) = ((pr, pc, pt) , (pr, pc, pt) )
= (pr, pc, pt) = pA
Identity Matrices. There is a special group of matrices called the identity matrices. These are square matrices with the property that they have 1's on the main diagonal and 0's everywhere else. A square matrix is one where the number of rows and columns are equal. The main diagonal of a matrix A are those entries whose row and column subscripts are equal, i.e. the entries Aii for some i. The identity matrices are denoted by I. Here are some identity matrices.
I = = the 2(2 identity matrix
I = = the 3(3 identity matrix
I = = the n(n identity matrix
Thus
Iij =
These are called the identity matrices because they act like the number 1 for matrix multiplication. If x is a column vector and p is a row vector then
Ix = x
pI = p
To see the first of these two relations, consider the ith component of Ix.
(Ix)i =
Since Iij = 0 unless j = i one has (Ix)i = Iiixi = xi. So Ix = x.
Another Way to Multiply a Vector by a Matrix. The following proposition gives another way of viewing multiplication of a vector by a matrix.
Proposition 1.
a. Ax is the linear combination of the columns of A using the components of x as the coefficients, i.e.
Ax = x1A●,1 + x2A●,2 + ( + xnA●,n =
b. pA is the linear combination of the rows of A using the components of p as the coefficients, i.e.
pA = p1A1,● + p2A2,● + ( + pmAm,● =
Proof. To prove part a note that
()i = = = (Ax)i
The proof of part b is similar. //
Example 3.
Ax = = 2 + 3 = + =
pA = (2, 3, 5) = 2(5, 7) + 3(2, 3) + 5(3, 5)
= (10, 14) + (6, 9) + (15, 25) = (31, 48)
Algebraic Properties of Multiplication. The product of a vector and a matrix satisfies many of the familiar algebraic properties of multiplication.
Proposition 2. If A and B are matrices, x and y are column vectors, p and q are row vectors and t is a number then the following are true.
(A + B)x = Ax + Bx A(x + y) = Ax + Ay
p(A + B) = pA + pB (p + q)A = pA + qA
A(tx) = t(Ax) (tA)x = t(Ax)
(tp)A = t(pA) p(tA) = t(pA)
(Ax)T = xTAT (pA)T = ATpT
p(Ax) = (pA)x
Proof. These are all easy to prove, so we leave the proof of most of them for an exercise. We prove (Ax)T = xTAT and p(Ax) = p(Ax) as illustrations. To prove the first note that ((Ax)T)j = (Ax)j = (Aj,●)x = xT(Aj,●)T = xT((AT)●,j) = (xTAT)j. Note that we used the fact that the jth column of AT is the same as the transpose of the jth row of A. To prove p(Ax) = p(Ax) note that
p(Ax) = = = =
(pA)x = = = =
where bij = piAijxj. In general = because in both case one is summing bij over all combinations of i and j where i runs from 1 to n and j runs from one to n. //
The following proposition is the analogue of Proposition 2 in section 1.4. It says that multiplication by a matrix gives a linear function. It also says that all linear functions z = T(x) where x and z are column vectors can be obtained by multiplication by a matrix.
Propostion 3. Let A be an m(n matrix and let z = T(x) = Ax for any column vector x = with n components. Then T is linear, i.e. T(x + y) = T(x) + T(y) and T(tx) = tT(x) for any vectors x and y and number t. Furthermore, if z = T(x) is a linear function that maps vectors x = to vectors z = with m components then there is an m(n matrix A such that T(x) = Ax.
Proof. The proof is very similar to the proof of Proposition 2 in section 1.4. First suppose that z = T(x)= Ax for any x. Then T(x + y) = A(x + y) = Ax + Ay = T(x) + T(y) where we used Proposition 2 for the second equality. Similarly T(tx) = A(tx) = t(Ax) = tT(x) where we again used Proposition 2 for the second equality. So T is linear.
Now suppose z = T(x) is linear. We can write x = = x1e1 + x2e2 + … + xnen where ei = is the vector such that every component is zero except the ith. So T(x) = T(x1e1 + x2e2 + … + xnen) = x1T(e1) + x2T(e2) + … + xnT(en). If we let A be the matrix such that A●,j = T(ej), then T(x) = A●,1x1 + A●,2x2 + … + A●,nxn = Ax which is what we wanted to show. //
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