Lecture 17: Linearity of Soln set. Practical
applications.
Aim Lecture In lecture 13, saw solns to
lin eqns in 3 real var forms either i) ii) pt iii)
line iv) plane v) all R3. This lecture examines
Also look at a practical applicn of lin eqns.
Parametric
Lecture 3: Fields or Systems of Numbers
Q In what sense is C with addn
Aim Lecture Answer this question by
Fields: Let F be a set.
E.g.
Imprecise Defn F is a eld (technical name
for system of numbers) if
More precisely, suppose
i) an addn rule on F assign
Lecture 6:Solving polynomial equations
Aim Lecture The use of polar forms makes
it
Square Roots
e.g.1 Solve z 2 = 5 + 12i
A Let z = a+
Equate real & imag parts
Solve simultaneously by guessing or elim a
or better still
1
|z 2|
Quadratic eqn Quadratic form
Lecture 2: Arithmetic of Complex
Numbers
Aim of Lecture: Enlarge set of
Q What is
Possible A A real number
Motivation for complex numbers
Q Suppose graph of a polynomial fn is
e.g. y = x2 + bx + c
1
Whats d?
A
In e.g. quadratic formula =
Problem: If
Howev
Lecture 5: Eulers formula & applications
Aim Lecture Multn & computing powers/roots best
Lemma (cos + i sin )(cos
Proof LHS =
remark Here, mult numbers correponds to
1
Geom Interpretn
z = cos + i sin , w = r(
zw is w
Eulers Formula For
More gen,
ea+bi :=
Lecture 4: Argand Diagram. Polar Forms
Aim Lecture Though complex numbers
dont represent quantities they can
Argand Diagram
Represent the complex number z =
e.g. i has coords
Geom interpretation of addn
e.g. z = 1 + i, w = 2 + i.
1
contd
See z + w corresp
Lecture 7: Trigonometric identities from
complex numbers
Aim Lecture Eulers formula suggests
Binomial Formula
Defn-Propn For n , 0
the binomial
Its the no. ways of picking
Binomial Thm
(a + b)n =
1
Why?
Facts 1.ei =
2. sin =
3. cos =
Proof From picture
2
Lecture 8: Complex polynomials
Aim Lecture There is a rich theory associated to complex polynomials because
Defn A complex polynomial is an expression
Let C[z] denote
If all ai
Dene the degree
To a complex (or real) polynomial we obtain
1
a function p :
Lecture 12: Planes in Rn
Aim Lecture Describe & understand
Linear Combinations We start with some
motivational discussion.
1-Dim case: Given line x =
All possible dirn
2-Dim case: What are all possible
i.e. vectors whose head &
1
Suppose v, w = 0 lie on p
Lecture 11: Lines in Rn
Aim Lecture Describe & understand
Parallel 2 non-zero vectors v, w Rn are
parallel if
If v, w R3 or R2 this means corresponding geom vectors have same
Line Segments Put coord system on 3dim (or 2-dim) space.
1
Rem We often confuse
Lecture 10: Applications. Laws of
Arithmetic for Vectors
Aim of Lecture a) Well formulate some
practical problems using n-tuples.
b) Field axioms gives basic laws for simplifying manipulating numbers.
Let F = eld e.g.
Defn Given v, w Fn, dene the negative
Lecture 9: Geometric & Coordinate Vectors
Aim Lecture In nature, sometimes only
interested in the magnitude of a quantity e.g.
mass, area. Sometimes, also interested
Defn A geometric vector in 3-dim (or 2dim) space, usually denoted by bold symbol
like v (
Lecture 13: Systems of Linear Eqns
Aim Lecture Solns to many physical problems can be obtained by solving systems of
linear equations. We set up a general framework
Some simple examples you know
e.g.1 x y = 1
no. solns
corresponding to
e.g.2
xy =1
2x y =
Lecture 15: Gaussian Elimination. Reduced
Row Echelon Form.
Aim Lecture Recall (A|b) easy to solve if
A in row echelon form & EROs dont change
soln set. Show can always apply
This algorithm called
Gaussian Elimination
E.g. 1 Solve
0
1 1 2 0
1 1 0 0 1
(A|
Lecture 19: Transpose. Elementary
Matrices.
Aim Lecture Examine how transposes allow you to swap results from left
Propn 1 (A)T )T =
Proof Clear from any e.g.
Propn 2 For matrices A, B & F, the
following hold if they make sense.
1) (A + B)T =
2) (A)T =
3)
Lecture 18: Algebra of Matrices
Aim Lecture You can add & multiply
There are some important dierences however.
Notn-Defn Let Mmn(F) denote the set of
all
In Mmn we always have the zero matrix 0
which
e.g. in M23 zero matrix is
The (i, j)-th entry of A Mmn
Lecture 14: Row Echelon Form. Elementary
Row Operations.
Aim Lecture Key to solving lin eqns involves 2 concepts.
i) Elementary Row Operations (EROs): which
& ii) Row Echelon Form: which are systems
of lin eqns, suciently
Analyse Easy Example Solve
x 2y =
Lecture 16: Geometric Applications. Case of
Unique Solns
Aim Lecture Solve some problems arising
from chapter 2 material using
e.g. 1 a) Do the two lines x = (0, 3, 4) +
(1, 0, 2), x = (5, 1, 0) + (2, 1, 1) intersect
& b) if so, nd their pt of intersectio
Lecture 20: Inverse of a Matrix
Aim Lecture The notion of inverse allows
you to (left & right) divide & so is useful
Defn Let A Mmn. A left (resp. right)
inverse for A is a matrix B
If B is both a left & right we say A is invertible & B
N.B. If B is a lef
Lecture 22: The Dot Product of Vectors
Aim Lecture The dot product contains
info about
NOTE In this nal chapter, F = R always.
Length Recall from lecture 9, that for v =
(v1, . . . , vn)T Rn, the length
Defn made to extend defn for geometric vectors. Many
Lecture 26: Distances between lines, points
& planes
Aim Lecture The notions of projection
and cross product are useful for
Distance from point to plane Let P
be point in R3. Let V be plane containing pt
A with normal vector n.
Drop a perpendicular from P
Lecture 25: Planes in R3.
Aim Lecture Planes in R3 are determined
by a pt on them &
This yields
Cross product connects this with the
Point Normal Form Let V R3 be a
plane
Let c be (coord vector of) pt on V & n
Then coords of any pt x on V satisfy
1
()
by
Lecture 24: Cross Products.
Aim Lecture The study of 3-dim geom is
greatly aided by the notion
Warning Cross products only work in
Let v = (v1, v2, v3), w = (w1, w2, w3) R3
Defn Dene the cross product of
fake
vw =
real
=
N.B. This product is a vector in
I
Lecture 23: Orthogonality & Projections
Aim Lecture The dot product can be used
to study
Defn 2 vectors v, w Rn are orthogonal
N.B. If v, w non-zero this means the angle
We also say in this case that v, w are normal
or
Pythagoras Thm (in Rn)
Let A, B, C b
Lecture 21: Determinant of a Square Matrix
Aim Lecture Intro determinant of a square
matrix & examine
Apology Most proofs omitted this lecture.
Defn Let A Mmn. We dene by induction on n the number det A or |A| called
n = 1: A = (a)
n = 2: A =
Fact See lat
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