Lecture 3
Linear Equations and Matrices
•linear functions
•linear equations
•solving linear equations
3–1
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solving linear equations. 3–1 ... so, matrix multiplication is a linear function ... n-vector, can be expressed as y = Ax for some m × n matrix A.
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2022/2023
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Lecture 3
Linear Equations and Matrices
•^
linear functions
-^
linear equations
-^
solving linear equations
Linear functions
function
f
maps
n
-vectors into
m
-vectors is
linear
if it satisfies:
•^
scaling
: for any
n
-vector
x
, any scalar
α
,^
f^ (
αx
αf
(x
•^
superposition
: for any
n
-vectors
u
and
v
,^
f^ (
u^
v
f
(u
f
(v
example:
f
(x
y
, where
x
x^1 x^2 x^3
,^
y^
[^
x^3
x^1
3 x
1
x^2
]
let’s check scaling property:
f^ (
αx
[^
(αx
αx
αx
αx
]
α
[^
x^3
x^1
3 x
1
x^2
]
αf
(x
Linear Equations and Matrices
Composition of linear functions
suppose^ •
m
-vector
y
is a linear function of
n
-vector
x
,^
i.e.
,^
y^
Ax
where
A
is
m
×
n
•^
p-vector
z
is a linear function of
y
,^ i.e.
,^
z^
By
where
B
is
p
×
m
then
z
is a linear function of
x
, and
z
By
BA
)x
so
matrix multiplication
corresponds to
composition
of linear functions,
i.e.
, linear functions of linear functions of some variables Linear Equations and Matrices
Linear equations
an equation in the variables
x
,... , x 1
n^
is called
linear
if each side consists
of a sum of multiples of
x
, and a constant,i
e.g.
x
2
x
3
x^1
is a linear equation in
x
, x 1
, x 2
3
any set of
m
linear equations in the variables
x
,... , x 1
n^
can be
represented by the compact matrix equation
Ax
b,
where
A
is an
m
×
n
matrix and
b
is an
m
-vector
Linear Equations and Matrices
step 2:
rewrite equations as a single matrix equation:
[^
]
x (^1) x (^2) x 3
[^
]
•^
ith row of
A
gives the coefficients of the
i
th equation
•^
jth column of
A
gives the coefficients of
x
j^
in the equations
•^
ith entry of
b
gives the constant in the
i
th equation
Linear Equations and Matrices
Solving linear equations
suppose we have
n
linear equations in
n
variables
x
,... , x 1
n
let’s write it in compact matrix form as
Ax
b
, where
A
is an
n
×
n
matrix, and
b
is an
n
-vector
suppose
A
is invertible,
i.e.
, its inverse
A
−
1
exists
multiply both sides of
Ax
b
on the left by
A
−
A
−
Ax
A
−
1 b.
lefthand side simplifies to
A
−
1 Ax
Ix
x
, so we’ve solved the linear
equations:
x
A
−^1
b
Linear Equations and Matrices
when
A
isn’t invertible,
i.e.
, inverse doesn’t exist,
•^
one or more of the equations is redundant(i.e.
, can be obtained from the others)
•^
the equations are inconsistent or contradictory (these facts are studied in linear algebra)in practice:
A
isn’t invertible means you’ve set up the wrong equations, or
don’t have enough of them Linear Equations and Matrices
Solving linear equations in practice
to solve
Ax
b
i.e.
, compute
x
A
−
1 b
) by computer, we don’t compute
A
−
1 , then multiply it by
b
(but that would work!)
practical methods compute
x
A
−
1 b
directly, via specialized methods
(studied in numerical linear algebra)standard methods, that work for any (invertible)
A
, require about
n
3
multiplies & adds to compute
x
A
−
1 b
but modern computers are very fast, so solving say a set of
equations
in
variables takes only a second or so, even on a small computer
... which is simply
amazing
Linear Equations and Matrices