Download Learning Objectives 1. Describe a system of linear (scalar

Learning Objectives
1. Describe a system of linear (scalar) equations geometrically (when there are up to n = 3 variables).
Describe the geometry of the system when there is
(a) exactly one solution (a unique solution),
(b) infinitely many solutions,
(c) or no solution.
2. Interpret a linear system of equations as a matrix-vector equation, and explain why it is useful.
3. Use row operations and augmented matrices to determine whether a system of algebraic equations has
(a) exactly one solution (a unique solution),
(b) infinitely many solutions,
(c) or no solution.
4. Use determinants to state whether a solution to a system of linear equations is unique.
5. Define what an nth order system of ODEs is, and describe why they are relevant.
6. Convert nth order ODEs to a system of n first-order ODEs.
7. Solve an nth order linear homogeneous system of the form ~x˙ = A~x, for constant A.
~
8. Solve an nth order linear nonhomogeneous system of the form ~x˙ = A~x + E(t),
for constant A.
Objective 1
Based on the observation that x + y = 1 is a line in the xy-plane. Similarly, the equation x − 2y + 3z = 1 is
a plane in R3 that has normal vector h1, −2, 3i and contains the point (1,0,0)
Objective 2
Definition: Consider the m scalar equations in the n scalar variables c1 , c2 , . . . , cn given below:
a11 c1 + a12 c2 + a13 c3 + · · · + a1n cn = b1
a21 c1 + a22 c2 + a23 c3 + · · · + a2n cn = b2
a31 c1 + a32 c2 + a33 c3 + · · · + a3n cn = b3
..
.
(1)
am1 c1 + am2 c2 + am3 + · · · + amn cn = bm
where bi for 1 ≤ i ≤ m and the coefficients aij for 1 ≤ i ≤ m and 1 ≤ j ≤ n are real or complex numbers.
We write these m scalar equations in n variables as a matrix-vector equation of the form A~x = ~b where
1. A is an m × n matrix of m rows and n columns. The entry in row i, column j is
aij : the expression multiplying the j th unknown in equation i
That is,

a11
 a21


A =  a31
 ..
 .
a12
a22
a32
..
.
a13
a23
a33
..
.
···
···
···
..
.

a1n
a2n 

a3n 
,
.. 
. 
am1
am2
am3
···
amn
2. ~b is a m × 1 column vector,


b1
 b2 
 

~b = 
 b3  ,
 .. 
 . 
bm
1
where bi is the term that is independent of the unknowns in equation i,
3. and ~x is a n × 1 column vector of the unknowns. That is,
 
c1
 c2 
 
 
~x =  c3  ,
 .. 
.
cn
where cj is the j th unknown.
So, the matrix vector equation Ax = b

a11
 a21

 a31

 ..
 .
am1
is the same as writing
   
b1
c1
a12 a13 · · · a1n
 c2   b2 
a22 a23 · · · a2n 
   
   
a32 a33 · · · a3n 
  c3  =  b3 
..
..
..
..   ..   .. 
.
.
.
.  .   . 
am2
am3
···
amn
cn
(2)
bm
Definition: The augmented matrix associated with the system (1) is the horizontal
A and b. That is

 
a11 a12 a13 · · · a1n b1
a11 a12 a13 · · · a1n
 a21 a22 a23 · · · a2n b2   a21 a22 a23 · · · a2n

 

 
[A|b] =  a31 a32 a33 · · · a3n b3  =  a31 a32 a33 · · · a3n
 ..
  ..
..
..
..
..
..
..
..
..
 .
  .
.
.
.
.
.
.
.
.
am1 am2 am3 · · · amn bm
am1 am2 am3 · · · amn
concatenation of

b1
b2 

b3 



bm
(3)
Note the vertical line separating the A and b is just a visual cue that the resulting m × (n + 1) matrix is
associated with a system of algebraic equations, hence the last equality above.
Objective 3
Definition: A sequence of any of the following three row operations performed on a matrix M yields a
row equivalent matrix:
1. (Replacement) Replace row i with a sum of row i and a multiple of row k.
2. (Scaling) Multiply all entries in row i by a nonzero constant.
3. (Interchange) Swap two rows.
Theorem: If the augmented matrices of two linear systems are row equivalent, then the two systems have
the same solution.
Definition:
A matrix is in echelon form provided
1. All nonzero rows are above any rows of all zeros.
2. The left-most nonzero entry of each row is in a column to the right of the left-most nonzero entry
in the row above it. We call the location of these entries in the matrix pivot positions. A column
containing a pivot position is called a pivot column. The entry in a pivot position is referred to as a
pivot.
3. All entries in a column below the entry in a pivot position are zero.
A matrix is in reduced row echelon form provided it is in echelon form, as well as
4. The entry in each pivot position is one.
5. The entry in each pivot is the only nonzero entry in its column.
2
Theorem: The reduced row echelon form of a matrix is unique. That is, each matrix is row equivalent to
one and only one reduced echelon matrix.
Objective 4 Definition: Let A be a n × n matrix. The determinant of A is
det(A) =
=
n
X
j=1
n
X
aij Cij
for any i
aij Cij
for any j,
i=1
where Cij is the (i, j)-cofactor of A, defined by Cij = (−1)i+j det Aij , and Aij is the (n − 1) × (n − 1)
submatrix of A that results when ignoring column i, row j of A.
Theorem: Let A be a n × n, then A~x = ~b has a unique solution for all ~b if and only if det(A) 6= 0.
Objective 5
Definition: In general, a n first-order ODEs can be written as
dx1
= f1 x1 (t), x2 (t), . . . , xn (t)
dt
dx2
= f2 x1 (t), x2 (t), . . . , xn (t)
dt
..
.
dxn
= fn x1 (t), x2 (t), . . . , xn (t)
dt
An nth -order linear system of ODEs is when fj has the form:
dx1
= a11 (t)x1 (t) + a12 (t)x2 (t) + a13 (t)x3 (t) + · · · + a1n (t)xn (t) + E1 (t)
dt
dx2
= a21 (t)x1 (t) + a22 (t)x2 (t) + a23 (t)x3 (t) + · · · + a2n (t)xn (t) + E2 (t)
dt
dx2
= a31 (t)x1 (t) + a32 (t)x2 (t) + a33 (t)x3 (t) + · · · + a3n (t)xn (t) + E3 (t)
dt
..
.
dxn
= an1 (t)x1 (t) + an2 (t)x2 (t) + an3 (t)x3 (t) + · · · + ann (t)xn (t) + En (t)
dt
We write (4) in matrix-vector form as
~
~
D~x = A(t)~x + E(t),
or equivalently ~x˙ = A(t)~x + E(t),
where

a11 (t)
 a21 (t)

A(t) =  .
 ..
a12 (t)
a22 (t)
..
.
···
···
..
.
an1 (t) an2 (t) · · ·

a1n (t)
a2n (t) 

.. 
. 
ann (t)


x1 (t)
 x2 (t) 


~x =  . 
.
 . 
xn (t)
3


E1 (t)
 E2 (t) 


~
E(t)
= . 
.
 . 
En (t)
(4)
Objective 6 Consider an nth order line ODE
L(x(t)) = E(t).
The following steps allow us to write the ODE in phase variable form:
1. Introduce n (same as the order) phase variables as the unknown x(t) and its derivatives:
x1 (t) = x(t)
x2 (t) = x0 (t)
x3 (t) = x00 (t)
..
.
xn−1 (t) = x(n−2) (t)
xn (t) = x(n−1) (t)
2. Differentiate both sides (with respect to t) each of the n above equations:
x01 (t) = x0 (t)
x02 (t) = x00 (t)
x03 (t) = x000 (t)
..
.
x0n−1 (t) = x(n−1) (t)
x0n (t) = x(n) (t)
3. Rewrite the right-hand-side in terms of phase variables (no derivatives):
x01 (t) = x2 (t)
x02 (t) = x3 (t)
x03 (t) = x4 (t)
..
.
x0n−1 (t) = xn (t)
x0n (t) = x(n) (t)
The boxed term above can be rewritten in terms of the phase variables using the ODE L(x) = E.
4