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MA 106 Linear Algebra: Introduction

U. K. AnandavardhananIIT Bombay

January 07, 2013

U. K. Anandavardhanan IIT Bombay MA 106 Linear Algebra: Introduction

Welcome!

Welcome aboard MA 106!

We’ll cover Linear Algebra in the next six weeks or so. Please referto the information booklet for all the details regarding the flight.We expect a smooth journey, however keep your seatbelts fastenedthroughout, and not just when the fasten-the-seatbelt sign isilluminated (Feb 06 & Feb 18). This is a non-bunking flight;attendance in the lectures and tutorials is absolutely compulsory.In case you need any further assistance, please get in touch withme or one of your course associates; we are here to help you.


Linear Maps

MA 106 is the study of “vector spaces” and the “right” mapsbetween them. For now, keep Rn as an example of a vector space.Essentially it means: you can add two vectors, and what you get isa vector; you can multiply a vector by a scalar, and what you get isa vector. What are the right maps between vector spaces? Theseshould respect both addition and scalar multiplication; i.e.,

T (v + w) = T (v) + T (w),

T (c · v) = c · T (v).

T : R→ R given byT (x) = x2

is not linear (why?);T (x) = cx ,

for a constant c , is linear (why?).


Linear Maps

What are the linear maps from R2 to R? If T : R2 → R is linear,then,

T ((x , y)) = T (x · (1, 0) + y · (0, 1))

= x · T ((1, 0)) + y · T ((0, 1))

= ax + by ,

where a = T ((1, 0)) and b = T ((0, 1)). Note that this map isindeed linear. Thus, any linear map from R2 to R is necessarily ofthe form

T ((x , y)) = ax + by ,

for some a, b ∈ R.


Linear Maps

What about linear maps from R2 to R3? Such a T takes a pair(x , y) to a triple. How do you want to denote this triple?

(T1(x , y),T2(x , y),T3(x , y)),

where T1,T2,T3 are maps from R2 to R. Are these linear maps?Yes (why?). But the Ti ’s are given by

Ti ((x , y)) = aix + biy ,

for some ai , bi ∈ R. Is there a better way of writing? Yes, a totallyawesome way!


Linear Maps

Write the row vector (x , y) as the column vector[xy

],

and write the row vector (T1(x , y),T2(x , y),T3(x , y)) as thecolumn vector T1(x , y)

T2(x , y)T3(x , y)

,

and ask the question: what takes

[xy

]7→

a1x + b1ya2x + b2ya3x + b3y

?


Linear Maps

Answer: a1 b1

a2 b2

a3 b3

· [ xy

]=

a1x + b1ya2x + b2ya3x + b3y

.

Remark: We prefer to multiply a vector by a matrix on the left.This is just our convention. If you want to multiply on the right,the matrix form would be[

x y]·[

a1 a2 a3b1 b2 b3

]=[

a1x + b1y a2x + b2y a3x + b3y].


Linear Maps

Thus, a linear transformation

T : Rn → Rm

can be represented by

an m × n matrix,

and conversely, an m × n matrix does give rise to a lineartransformation T : Rn → Rm.

Remark: The above representation does depend on the fact thatwe’ve expanded a pair along {(1, 0), (0, 1)} on the left and a triplealong {(1, 0, 0), (0, 1, 0), (0, 0, 1)} on the right. An importantquestion to keep in mind: what’s so special about these choices ofvectors? We’ll return to this theme in a major way.


Matrices

How do you usually denote the entries of an m × n matrix? Theaadhaar of an entry is its row and column information; the entry atrow i , column j is denoted by aij .

Recall:

1 matrix addition: (aij)m×n + (bij)m×n = (aij + bij)m×n.2 scalar multiplication: c · (aij)m×n = (caij)m×n .

3 matrix multiplication: (aij)m×p · (bij)p×n = (

p∑k=1

aikbkj)m×n.

4 transpose: t(aij)m×n = (bji )n×m, where bji = aij .

Recall:

+ additive identity (X), additive inverse (X), associativity of +(X), commutativity of + (X)

+ mult. identity (X), mult. inverse (?), associativity of × (X),commutativity of × (7)

+ distributive property (X)


Matrices

Remark: Multiplicative inverse is defined only for a square matrix,and this need not exist. For instance,[

a bc d

]·[

1 11 1

]6=[

1 00 1

]for any choice of a, b, c, d ∈ R. However, if it exists, it is unique.Indeed, if AB = BA = I and AC = CA = I , then

B = B · I = B(AC ) = (BA)C = I · C = C .

Remark: We’ll prove later that AB = I automatically impliesBA = I .

Exercise: Prove that t(AB) = tB · tA and t(A−1) = (tA)−1.


“the” math course

When you try to model the real world, the modeling functions thatwe get may be quite complicated; the shape they define may beintricate - a curve may be curved very funnily! Assume they are atleast differentiable as in MA 105. Do we understand the behaviourof such a function, a curve, a surface? MA 105 told you that youcould understand the “local” behaviour around a given point fairlyeasily. A curve (or a surface) may be difficult, but at a point, thetangent line (or the tangent plane) tells us a lot about thebehaviour of the curve (or the surface) at that point. If adifferentiable f : U ⊂ Rn → Rm is given, at a point a ∈ U, thetotal derivative Df (a) contains all the local information; i.e.,information around a small neighbourhood of a. But what isDf (a)?

Df (a) =

(∂fi∂xj

)m×n

,

a linear transformation from Rn to Rm.


The punch line is . . .

locally, MA 105 is just MA 106!

You’ll see in the next half-sem that MA 108 rests on MA 106 aswell.


And MA 106 makes you rich!


Paging for passengers . . .

1 120010047 VALLABHANENI JYOTHI SRIRAM2 120070001 SUVIGYA S CHANDRA3 120010014 HATKAR SHANTANU SUBHASH4 12D070038 LAKSHAY KUMAR5 120010040 AMAN VERMA6 12D070063 JAGRITI SINGH7 12D070008 PAI SAGUN GURUDAS8 12D070036 KRATI SETHI9 120070005 SANKAGIRI SURYANARAYANA

10 12D070055 SRINIVAS BEHERA11 120010010 DARSH JAIDIP SHAH12 120070026 SITU KUMAR GUPTA13 120070003 RIDDHISH BHALODIA14 12D070051 ASHISH GOYAL15 12D070012 ADITYA NAMBIAR16 120070036 BAVISETTY VENKATA SAI NARAYANA17 12D070021 PASAD ANKITA JITESH18 12D070023 PATIL ASHLESHA HARSHADIP19 12D070029 NIKLESH LALWANI20 12D070026 ABHISHEK VISHWA



Let A,B,C ,D,E be 5 webpages. Let’s track the links and formthe array:

A B C D EA 0 1/2 1/2 1/4 0 1.25B 1/3 0 1/2 1/4 0 1.08C 0 0 0 1/4 1 1.25D 1/3 1/2 0 0 0 0.83E 1/3 0 0 1/4 0 0.58

If every webpage had the same weight, the ranking isA = C > B > D > E .



But in real life, you value a link more if it is from a moreprestigious webpage. Thus, a link from a more prestigious pageshould have more of a weight in the rank computation. But then,how to determine which one is a more prestigious page?

Let x1, x2, x3, x4, x5 be the weights of A,B,C ,D,E .

A B C D EA 0 1/2 1/2 1/4 0 1.25 1

2x2 + 12x3 + 1

4x4B 1/3 0 1/2 1/4 0 1.08 1

3x1 + 12x3 + 1

4x4C 0 0 0 1/4 1 1.25 1

4x4 + x5D 1/3 1/2 0 0 0 0.83 1

3x1 + 12x2

E 1/3 0 0 1/4 0 0.58 13x1 + 1

4x4

If we know the weights, we know the weighted ranking! But wedon’t know the weights! Any ideas?



Declare the weighted points that a page gets as its weight! We get:

1

2x2 +

1

2x3 +

1

4x4 = x1

1

3x1 +

1

2x3 +

1

4x4 = x2

1

4x4 + x5 = x3

1

3x1 +

1

2x2 = x4

1

3x1 +

1

4x4 = x5

Questions to ask:

Does this system have a solution (non-trivial)?

If Yes, is it unique (up to scalars)?

(Why should we ask these questions?)



In other words, is there a unique (up to scalars) non-trivial solutionto

0 1/2 1/2 1/4 01/3 0 1/2 1/4 0

0 0 0 1/4 11/3 1/2 0 0 01/3 0 0 1/4 0

·

x1x2x3x4x5

=

x1x2x3x4x5

?



A unique solution is guaranteed by results from MA 106 providedyou “perturb” the system a little bit using a “damping factor”. Ifwe follow Google’s algorithm, we get:

A B C D EA 0 1/2 1/2 1/4 0 1.25 1.2395766185B 1/3 0 1/2 1/4 0 1.08 1.1163137447C 0 0 0 1/4 1 1.25 0.9597953736D 1/3 1/2 0 0 0 0.83 0.9756351179E 1/3 0 0 1/4 0 0.58 0.7086791453

Notice how C slips from 1st to 4th; indeed the full 1 point that Cgets from E does not count much because E doesn’t carry muchweight!


From tomorrow onwards . . .

. . . we’ll systematically start attacking the question of solvingsystems of linear equations. Keep the seatbelts fastened, and

sit back, relax, and enjoy the flight!


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