Lecture 7: Basis and Coordinates
Aim of Lecture: Develop notion of coordinate systems and the concept of bases.
V = vector space/ eld F
Defn A subset B V is a basis for V if
a.
& b.
E.g. 1 V = Pk has standard basis
B=
Check: Span B =
B is lin indep by
If
Lecture 14: Invertible Linear Maps
Aim Lecture Coordinates allow you to
identify n dim vector spaces with
More gen, invertible
Defn (One-to-one) Let X, Y be sets & f :
X Y be a function. We say that f is
one-to-one (1-1) or injective if the soln
i.e. f (x
Lecture 16: Intro to Probability & Random
Variables
Aim Lecture Introduce some of the language of statistics.
E.g. 1 In statistics one might wish to examine:
a) trivia scores of
b) no. times someone has
c) high temperature
In each case have
1. A populatio
Lecture 9: Existence & Construction of
Bases
Aim We show the existence of bases in special cases and illustrate methods of nding
them.
Thm 1 Let V = vector space/ eld F
Let S V be a spanning set of n elts. Any
subspace W
Proof: Omitted. We will prove the
Lecture 13: Solving Linear Eqns. Image
Aim Lecture For T : V W linear,
understand when you can solve
Image
Defn-Propn (Image) Let T : V W
be linear. The image of T is
Also rank T :=
If T = TA also write
Note im A = col A.
Proof: Follows easily from subspa
Lecture 8: Basis and Dimension
More Examples of Bases
From thm lecture 6 we know
Thm 1 Let V = vector space/ eld F.
B = cfw_v1, . . . , vn is a basis i every v V
is uniquely
E.g. 1 V = Mmn(F). For i
dene Eij =
i.e. 0 everywhere but
1
B = cfw_Eij is a ba
Lecture 11: Geom Examples of Lin Maps
Aim Lecture 11 Exhibit various geometric transformations such as rotations and reections as linear maps.
Orthogonal Projections
Let u Rn be
Let T : Rn Rn be projn onto line L :=
Recall T v = (v .
T is linear.
Why? Add
Lecture 10: Linear Maps or Transformations
Recall, a (homogeneous) linear function f :
Rn R is one of the form
f (x1, . . . , xn)T =
More generally, a lin fn f : Rn Rm is
one of the form
Aim Lecture 10 Generalise the notion of
linear functions to linear m
Lecture 12: Solving Linear Eqns in
Arbitrary Vector Spaces
E.g. 1 Multiplication by a function is linear. For a real-valued fn g(x) on R, dene
T : R[R] R[R] by
Then T is lin by the distrib &
E.g. 2 Dene T : P2(R) P3(R) by
dp
T p = (1 + x) 2p
dx
Note T is
Lecture 17: Mean and Variance for Discrete
Random Variables
Aim Lecture Dene and compute the mean
& variance for a discrete random var.
Recall for random var X on a population of
size n, with values xk occurring with freq fk
then,
x=
f
k
Since n
Defn (Exp
Lecture 19: Scaling. Normal Distribution
Aim Introduce the most important distribution, the normal distribution.
Probability Density of g(X)
X cont random var with
Let g(x) be increasing dible
Lemma Let Y = g(X) have prob density
f1(y) where y = g(x). The
Lecture 23: Intro to Eigenbases
Inevitable Woe
Q Why did we introduce abstract notion of
vector spaces?
A 1. To handle innite dim
2. Defn is
Let V = vector space/ eld F
Next few lectures, study lin maps of form
i.e. domain =
Aim T : V V often picks out it
Lecture 26: Powers of Matrices
Aim lecture: See applic of diag to computing powers of matrices & studying discretetime systems.
E.g. 1 Consider T = TA : R2 R2 with
e-vectors
Let B =
T 2 v1 =
1
T 3 v1 =
T k v1 =
Its thus easy to compute powers of T wrt
If
Revision: Summary & Sample Questions.
Ch 6: Vector Spaces
Main Point: Any nite dimensional vector
space looks just like Rn.
Q1 Is S = cfw_A Mnn(R)| det A = 0 a
subspace of Mnn?
Q2 Is
2
1
= AT
S = cfw_A M2(R)|A
1
1
a subspace of M22? If so nd a basis f
Lecture 24: Finding Eigenbases
Consider T : V
Examined last time certain preferred bases
for
E.g.
Study of T breaks down to studying
Q 1. How do you nd these preferred bases?
2. How do these bases improve our under1
standing of T & simplify
Aim this lect
Lecture 25: Diagonalisation & Di Eqns
Let T : V V have an e-basis cfw_v1, . . . , vn
& let Li = Span vi.
Aim lecture: See how study of T decomposes into study of
E.g. 1 y1(t) = popn of hobbits in
y2(t) = popn of orcs
If 2 popn kept separate as here then p
Lecture 18: Continuous Random Variables
A continuous random var X : R can
take any value in R.
E.g. X = humidity in Sydney.
Aim Lecture Set up probability theory for
these random var.
Approach
Consider e.g. above. Well use
Range of values is
Well divide t
Lecture 20: Normal Probabilities.
Independent Random Variables.
Aim Compute probabilities of normally distributed variables. Introduce the notion of independent random variables.
Tables
Consider normal prob distribn:
(z) =
(z) =
Note: Cant integrate (z) b
Lecture 21: Sums. Means. Central Limit
Thm
Aim Compute E(X), Var(X) for sums &
means of random var. Observe how all prob
distribn lead to the normal distribn.
Sums of 2 Random Var
Discussion in discrete case only
See later courses for cont case (need mult
Lecture 22: Central Limit Thm.
Approximations.
Aim Look at some practical applications of
the central limit thm to approx probabilities.
E.g. 1 X = weight of student
is a random var with mean = 70 & std
devn = 10.
15 students climb up Rap
which supports 1
Lecture 1: Vector Spaces
Aim of Todays Lecture:
Properties of Rn: Write V = Rn, F = R.
For any u, v, w
1. Closure Under Addition:
2. Associative Law of Addition:
3. Commutative Law of Addition:
4. Existence of Zero:
1
5. Existence of Negatives:
6. Closur
Lecture 4: More on Linear Combns and Span
Matrix Interpretn of Lin Combn &
Span in Fm
a11
a12
a1n
a
a
a
21
22
2n
a1 =
, a2 =
, . . . , an =
F
.
.
.
.
.
.
am1
am2
amn
Propn The lin combn
x1 a1 + . . . xn an = A x
Proof:
1
a11 . . .
Lecture 2: Subspaces
Aim of Lecture: Solns to linear eqns are typically points, lines planes etc in Rn.
These are special subsets e.g. they are linear, not curved. This lecture introduces
Description(Subspace) Let V be a vector
space / eld F. A subset W V
Lecture 6: Linear Dependence contd
Testing Lin Independence in R[R]
No systematic method.
E.g. 1 cfw_ex, e2x, e3x, . . . is lin indep
Why? Suppose 1 ex + . . . + n enx = 0
with
How does Span(S) vary with S?
Propn V = vector space/ eld F
Let S1 S2 be subse
Lecture 3: Linear Combinations and Span
Aim of Lectures 3/4: Recall that a nonzero direction vector v Rn determines a line
through the origin.
2 non-parallel vectors v, w determines a plane
through the origin.
We generalise these ideas to arbitrary
Linear
Lecture 5: Linear Dependence
Thought Experiment
Study Span(S) inductively by adding one
vector at a time to S.
Start with non-zero w1 Rm and line
Let w2 Rm.
Usually (A) Span(w1, w2)
unless (B)
1
Suppose in case (A) above. Let w3 Rm.
Usually (A) Span(w1, w