matrix P that produces the row exchange. It comes from
exchanging the rows of I: Permutation P = " 0 1 1 0# and
PA = " 0 1 1 0#"0 2 3 4# = " 3 4 0 2# . P has the same
effect on b, exchanging b1 and b2. The new system is PAx
= Pb. The unknowns u and v are
1) A = 1 1 1 1 2 2 1 2 3 = 1 0 0 1 1 0 1 1 1
1 1 1 0 1 1 0 0 1 = LU. From A to U there are
subtractions of rows. From U to A there are additions of
rows. Example 4. (when U is the identity and L is the
same as A) Lower triangular case A = 1 0 0 `21 1 0
0 0 1 0 1 0 1 0 0 and P23 = 1 0 0 0 0 1 0 1 0
and P23P13A = d e f 0 a b 0 0 c One more
point: The permutation P23P13 will do both row
exchanges at once: P13 acts first P23P13 = 1 0 0 0 0
1010001010100=001100
0 1 0 = P. If we had known, we could have mul
in the next section. The other matrix EFG is nicer. In that
order, the numbers 2 from E and 1 from F and G were
not disturbed. They went straight into the product. It is
the wrong order for elimination. But fortunately it is the
right order for reversing
can be found using whole columns as in equation (5).
Therefore Ax is a combination of the columns of A. The
coefficients are the components of x. To multiply A times
x in n dimensions, we need a notation for the individual
entries in A. The entry in the i
columns in the same plane Solvable only for b in that
plane u 1 2 3 +v 1 0 1 +w 1 3 4
= b. (4) For b = (2,5,7) this was possible; for b = (2,5,6)
it was not. The reason is that those three columns lie in a
plane. Then every combination is also in the pl
and this course. 1.2 The Geometry of Linear Equations
The way to understand this subject is by example. We
begin with two extremely humble equations, recognizing
that you could solve them without a course in linear
algebra. Nevertheless I hope you will gi
equations for x, y, z, t are you solving? 21. When equation
1 is added to equation 2, which of these are changed: the
planes in the row picture, the column picture, the
coefficient matrix, the solution? 22. If (a,b) is a multiple
of (c,d) with abcd 6= 0,
from abs(A(1,1) should be 0.5. 1.4 Matrix Notation and
Matrix Multiplication With our 3 by 3 example, we are
able to write out all the equations in full. We can list the
elimination steps, which subtract a multiple of one
equation from another and reach a
complete, at least in the forward direction: Triangular
system 2u + v + w = 5 8v 2w = 12 1w = 2. (3) This
system is solved backward, bottom to top. The last
equation gives w = 2. Substituting into the second
equation, we find v = 1. Then the first equatio
= d1 d2 . . . dn 1 u12/d1
u13/d1 . . . 1 u23/d2 . . . . . . . . . 1 . (9) In the
last example all pivots were di = 1. In that case D = I. But
that was very exceptional, and normally LU is different
from LDU (also written LDV). The triangular factorization
simple matrices. Give a 2 by 2 example of this important
rule for matrix multiplication. 20. The matrix that rotates
the x-y plane by an angle is A() = " cos sin sin
cos # . Verify that A(1)A(2) = A(1+2) from the
identities for cos(1+2) and sin(1+ 2). Wha
curve by its tangent line, fit the surface by a plane, and
the problem becomes linear. The power of this subject
comes when you have ten variables, or 1000 variables,
instead of two. You might think I am exaggerating to use
the word beautiful for a basic
. Twice the first row of A has been subtracted from the
second row. Matrix multiplication is consistent with the
row operations of elimination. We can write the result
either as E(Ax) = Eb, applying E to both sides of our
equation, or as (EA)x = Eb. The
are x1 = (1,1,1) and x2 = (0,1,1) and x3 = (0,0,1), solve Ax
= b when b = (3,5,8). Challenge problem: What is A? 53.
Find all matrices A = " a b c d# that satisfy A " 1 1 1 1# = "
1 1 1 1# A. 54. If you multiply a northwest matrix A and a
southeast matrix
What multiple of equation 2 should be subtracted from
equation 3? 2x 4y = 6 x + 5y = 0. After this elimination
step, solve the triangular system. If the right-hand side
changes to (6,0), what is the new solution? 4. What
multiple ` of equation 1 should be
answer is: With a full set of n pivots, there is only one
solution. The system is non singular, and it is solved by
forward elimination and back-substitution. But if a zero
appears in a pivot position, elimination has to stop
either temporarily or permane
multiply by A and then multiply by A1 , you are back
where you started: Inverse matrix If b = Ax then A 1 b =
x. 1.6 Inverses and Transposes 51 Thus A 1Ax = x. The
matrix A 1 times A is the identity matrix. Not all
matrices have inverses. An inverse is im
right-hand side as a fifth column (and omit writing u, v, w,
z until the solution at the end). 25. Apply elimination to
the system u + v + w = 2 3u + 3v w = 6 u v + w = 1.
When a zero arises in the pivot position, exchange that
equation for the one below
MTH203:Assignment-11
1. Verify that u(x, y) = a ln(x2 + y 2 ) + b satises Laplaces equation uxx + uyy = 0 and
determine a and b so that u satises the boundary condition u = 0 on the circle
x2 + y 2 = 1 and u = 3 on the circle x2 + y 2 = 4.
2. (a) Let u sa
MTH203: Assignment-7
1. Expand the following functions in terms of Legendre polynomials over [1, 1]:
0 if 1 x < 0
(i) f (x) = x3 + x + 1
(ii) f (x) =
(rst three nonzero
x if 0 x 1
terms)
2. Locate and classify the singular points for the following dierent
MTH203: Assignment-9
1. Find the eigen values and eigen functions of the following Strum-Liouville problems:
(i) y + y = 0,
y(0) = y (1) + y(1) = 0
(ii) (xy ) + x1 y = 0,
y(1) = y (e) = 0.
2. If p(x), q(x), r(x) are all greater than zero on (a, b), then p
MTH203:Assignment-10
1. In the following, by elimination of the arbitrary function F , nd rst order partial
dierential equation satised by z:
(i) z = xy + F (x2 + y 2 )
(ii) z = x + y + F (xy)
xy
(iii) z = F
z
x
(iv) z = F
y
(v) F (x z, y z) = 0
(vi) F (x
MTH203: Assignment-8
1. Let F (s) be the Laplace transform of f (t). Find the Laplace transform of f (at) (a >
0).
2. Find the Laplace transforms:
(a) [t], (b) tm cosh bt (m non-negative integers), (c) et sin at,
(d)
et sin at
sin t cosh t
, (e)
,
t
t
(f)
MTH203: Assignment-3
1. Solve the following linear rst order dierential equations using the method of variation
of parameters:
(i) xy 2y = x4
(ii) y + (cos x)y = sin x cos x
2. The equation
dy
= a(x)y 2 + b(x)y + b(x)
dx
is called Riccatis equation. If y
MTH203: Assignment-5
1. Verify that y = x2 sin x and y = 0 both are solutions of the initial value problem
x2 y 4xy + (x2 + 6)y = 0,
y(0) = y (0) = 0.
Does it contradict the uniqueness?
2. Find general solution of the following dierential equations given
MTH203: Assignment-4
1. Reduce the following second order dierential equations to rst order dierential equations and hence nd general solution
(i) xy = y
(ii) y = y 3 + y
(iii) yy + y 2 + 1 = 0 (iv) y 2y coth x = 0
2. Find the curve y = y(x) passing throu