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
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
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
10100010101
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 eliminatio
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 notat
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 l
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 co
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 solutio
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 subtr
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
= 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
fr
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
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
. 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
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 y
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 n
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
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 m
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 pi
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 =
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
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),
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
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
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
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 genera
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