Download Multiplication of Vectors and Linear Functions

1.4 Multiplication of Vectors and Linear Functions
The previous three sections were concerned with vectors and vector operations. The next
three sections look at multiplication of vectors and matrices and linear functions which
are another important ingredient in applications of linear algebra. Multiplication and
linear functions are closely related and we begin with the product of a row vector and a
column vector.
 x1 
x
Definition 1. If a = (a1, a2, …, an) is a row vector and x =  .2  is a column vector with
 
 xn 
the same number of components then the product ax of a and x is the sum of the products
of corresponding components, i.e.
(1)
 x1 
n
x
ax = (a1, a2, …, an)  .2  = a1x1 + a2x2 + … + anxn =  aixi
 
i=1
 xn 
Example 1.
 6
(2, -5, 4) -1 = (2)(6) + (-5)(-1) + (4)(0) = 12 + 5 + 0 = 17
 0
The importance of this type of product is it gives us a compact way to express linear
functions.
Example 2. The linear function w = 2x – 4y + 7z can be written either as
 x
w = (2, -4, 7)  y  = au
z
or
 2
w = (x, y, z)  -4  = vb
7
where
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a = (2, -4, 7)
 x
u = y
z
v = (x, y, z)
 2
b =  -4 
7
Depending on the situation, it may be more natural to write the variables as a column
vector and the coefficients as a row vector or vice versa. For example, often we make a
set of prices a row vector and a set of corresponding amounts a column vector.
Definition 2. A function z = (x1, x2, …, xn) of the variables x1, x2, …, xn is said to be
linear if it has the form
(2)
 x1 
n
x
…
z = (x1, x2, …, xn) = a1x1 + a2x2 + + anxn =  ajxj = (a1, a2, …, an)  .2 
 
j=1
 xn 
where a1, a2, …, an represent constants.
In other words, a linear function of a certain collection of variables is one where the
variables are multiplied by numbers and then added up or one that is equivalent to a
function of this form. Putting it another way, if we think of the variables x1, x2, …, xn as
 x1 
x
a vector x =  .2 , then a function z = (x) is linear if it has the form z = ax where
 
 xn 
a = (a1, a2, …, an). Any other function is called a non-linear function.
Example 3. The following are linear functions
y = 7x
w = 2x – 4y + 7z
y = 4x1 + 3x2 - 7x3 + 5x4
y = (x + 2)2 – x2 – 4
The last is linear because it is equivalent to y = 2x. The following are non-linear
functions
A = xy
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p =
8.3NT
V
y = 4x1 + 3(x2)2
y = sin(x1) + 3x2
y = 7x + 2
w = 2x – 4y + 7z - 3
In many other mathematics courses the last two functions would be called linear
functions. However, in this course they won’t. Instead they will be called
inhomogeneous linear functions because they are a linear function plus a constant.
Most functions are non-linear. However, linear functions occur so frequently in
applications that they are worth a lot of attention. Furthermore, in many situations one
can approximate a non-linear function by one or more linear ones.
Example 4. An electronics company makes 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
resistors
capacitors
transistors
ethernet card
5
2
3
sound card
7
3
5
Let
e
s
r
c
t
= # of ethernet cards the company makes in a certain day
= # of sound card the company makes in a certain day
= # of resistors needed to produce the e ethernet cards and s sound cards
= # of capacitors needed to produce the e ethernet cards and s sound cards
= # of transistors needed to produce the e ethernet cards and s sound cards
Then
r = (5
resistors
resistors
) (e ethernet cards) + (7
) (s sound cards)
ethernet card
sound card
or
r = 5e + 7s
So r is a linear function of e and s. Similarly,
c = 2e + 3s
t = 3e + 5s
are linear functions. In this same context let
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pr
pc
pr
pe
ps
=
=
=
=
=
price of a resistor
price of a capacitor
price of a transistor
cost of all the resistors, inductors and transistors in an ethernet card
cost of all the resistors, inductors and transistors in an sound card
Then
pe = 5pr + 2pe + 3pt
ps = 7pr + 3pe + 2pt
are again linear functions.
Algebraic properties of the vector product. The vector product (1) has some of the
familiar properties of regular products.
 x1 
x
Propostion 1. Let a = (a1, a2, …, an) and b = (b1, b2, …, bn) be row vectors and x =  .2 
 
 xn 
 y1 
y
and y =  .2  be column vectors with the same number of components and let t be a
 
 yn 
number. Then the following hold.
(3)
a(x + y) = ax + ay
(a distributive law)
(4)
(a + b)x = ax + bx
(another distributive law)
(5)
(ta)x = t(ax)
(an associative law)
(6)
a(tx) = t(ax)
(another associative law)
(7)
ax = xTaT
Proof. We prove (3) and leave the proof of the others for exercises. One has
n
n
n
n
n
i=1
i=1
i=1
i=1
i=1
a(x + y) =  ai(x + y)i =  ai(xi + yi) =  [(aixi) + (ayi)] =  aixi +  aiyi = ax + ay. //
An alternative characterization of linear functions. There is an alternative
characterization of linear functions that allows one to extend the concept of linear
functions to other vector spaces.
1.4 - 4
Propostion 2. Let a = (a1, a2, …, an) be a row vector with n components and let the
function z = T(x) be defined by z = T(x) = ax = a1x1 + a2x2 + … + anxn for any column
 x1 
x
vector x =  .2  with n components. Then the function T(x) has the following two
 
 xn 
properties.
(8)
T(x + y) = T(x) + T(y)
(9)
T(tx) = t T(x)
for any vectors x and y and number t. Furthermore, any function z = T(x) satisfying (8)
and (9) is a linear function in the sense of Definition 2.
Proof. First suppose that z = T(x) is linear. To show that (8) holds note that
T(x + y) = a(x + y) by definition. By Propostion 1 above one has a(x + y) = ax + ay.
However, by definition, the right side is equal to T(x) + T(y), so (8) holds. To show that
(9) holds note that T(tx) = a(tx) by definition. By Propostion 1 above one has a(tx) =
t(ax). However, by definition, the right side is equal to tT(x), so (9) holds.


Now suppose z = T(x) is a function satisfying (8) and (9). Let e =


i
0
0
.
0
1
0
.
0

 be the vector


 x1 
x
such that every component is zero except the ith. Note that x =  .2  can be written as the
 
 xn 
linear combination x = x1e1 + x2e2 + … + xnen, so T(x) = T(x1e1 + x2e2 + … + xnen). Since
z = T(x) satisfies (8) and (9) one has x1T(e1) + x2T(e2) + … + xnT(en). If we let ai = T(ei),
then T(x) = a1x1 + a2x2 + … + anxn, so z = T(x) is a linear function in the sense of
Definition 2. //
The properties (8) and (9) of functions that are linear in the sense of Definition 2, are
used to extend the concept of a linear function to general vectors.
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Definition 3. Let z = T(x) be a function that assigns numbers or vectors z to all vectors x
in some vector space. Then T is said to be linear if it satisfies (8) and (9), i.e.
T(x + y) = T(x) + T(y)
T(tx) = t T(x)
Proposition 2 shows that Definitions 2 and 3 agrees when x is a column vector. Often
books write Tx instead of T(x) if T is a linear function because T(x) acts like a product of
T and x.
Example 5. Let V be the vector space of all functions z = f(x) defined for 0  x  1 and
let  be a fixed point lying in the interval 0    1. If f is in V, let
1
Rf = f 2
Sf = f ()
1
1
3
5
7
Tf = 4 [ f 8 + f 8 + f 8 + f 8 ]
Then R, S and T are linear functions.
We prove R is linear; the proofs that S and T are linear are similar. One has
1
1
1
R(f + g) = (f + g)2 = f2 + g2 = Rf + Rg, so R satisfies (8), Also,
1
1
R(tf) = (tf)2 = t[f2] = t(Rf), so R satisfies (9). //
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Example 6. Let V be the vector space of all continuous functions z = f(x) defined for
0  x  1. If f is in V and 0  x  1, let
1
Rf = 
 f(x) dx
0
1
Sf = 
 f(x) sin(x) dx
0
x
(Tf)(x) = 
 f(t) dt
0
Then R and S are real valued linear functions. Furthermore and Tf is a continuous
function defined for 0  x  1 and T is a linear function from V to V.
We prove R is linear; the proofs that S and T are linear are similar. One has
1
1
1
1
R(f + g) = 
 (f + g)(x) dx = 
 (f(x) + g(x)) dx = 
 f(x) dx + 
 g(x) dx = Rf + Rg, so R satisfies (8),
0
0
1
0
1
0
1
Also, R(tf) = 
 (tf)(x) dx = 
 t(f(x)) dx = t 
 f(x) dx = t(Rf), so R satisfies (9). //
0
0
0
Example 7. Let V be the vector space of all functions z = f(x) defined for 0  x  1 which
df
have the property that the derivative f (x) = dx is also defined for 0  x  1. Let W be the
vector space of all functions z = f(x) defined for 0  x  1. If f is in V and 0  x  1, let
df
Tf = dx
Then T is a linear function from V to W.
d
df
dg
To prove this note that T(f + g) = dx (f + g) = dx + dx = Tf + Tg, so T satisfies (8), Also,
d
1
df
T(tf) = dx (tf) = 
 t(f(x)) dx = tdx, so T satisfies (9). //
0
1.4 - 7
Proposition 3. If V and W are vector spaces and S and T are linear functions from V to W
and t is a real number, then S + T and tT are also linear functions from V to W. Thus the
set of linear functions from V to W is a subspace of the vector space of all functions from
V to W.
Proof. We show S + T is linear; the proof that tT is linear is similar. One has
(S + T)(x + y) = S(x + y) + T(x + y) = Sx + Sy + Tx + Ty = Sx + Tx + Sy + Ty =
(S + T)x + (S + T)y, so S + T satisfies (8). Also, (S + T)(tx) = S(tx) + T(tx) = t(Sx) + t(Sy)
= t(Sx + Tx) = t[(S + T)x], so S + T satisfies (9). //
1.4 - 8