Transposes, vector subspaces - PowerPoint Presentation

Slide1

Transposes, vector subspaces

Daniel Svozil

based on excelent video lectures by Gilbert Strang, MIT

http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm

Lectur

e

5, Lecture 6Slide2

Transposes

How to write tra

n

spo

sition element-wise?(AT)ij = Aji

Symmetric matrices - no change upon transposing i.e. AT = A. Can you figure out some example?

RTR is always symmetric. Do it for the following R:

Why is R

T

R symmetric? Slide3

Vector spaces

What the hell are vector spaces and subspaces?

Which operations

can we do

with vectors? Suggestions?

add two vectors multiply vector by a number

To legitimately talk about a space of vectors, the requirement is that we should be able to add the things and multiply by numbers and that there should be some decent rules satisfied. Slide4

Let me say again what the word space means. It is a bunch of vectors, but not any bunch of vectors, it has to allow me to do the operations that vectors are for (add vectors and multiply by numbers, i.e. linear combination).

Of course, the results of such operations MUST lie also in that space!

Example: R

2

– two real numbers (i.e. two dimensional vectors)Give me some examples of such vectors and draw them in a graph (volunteer).So we could say that R2 is what?

xy planeSlide5

However the point is it's a vector space because all those vectors are in there.

Remove one of them – [0 0]. This is actually awful. Why?

Because I have to be allowed to multiply a vector by ANY scalar, including zero.

I also have to be able to add an opposite vector, and again I get an origin.

There is no vector life without origin !OK, R2 is an easy vector space, and so is R3.Slide6

Is line given as [3 2 0] in R

2 or in R3?

So what’s in R

n

?all vectors with n componentsWe can add two vectors, and we stay in Rn. We multiply any vector by a number, and we stay in Rn. Any linear combinations stays in Rn.

Propose some space where you can do addition, but it’s still not a vector space (i.e. you can’t do multiplication).Slide7

x

y

it’s not a vector space, because it is

not

closed

under multiplication

(however, it is closed under vector addition)

So a vector space has to be closed under multiplication and addition of vectors.

In other words, linear combinations.Slide8

Vector subspaces

R

n

is a good space, but raher large. We’ll be interested in spaces that are inside R

n.OK, so far we messed up R2, but it was unfortunate, we didn’t get a vector space.Now tell me a vector space that is part of R2 and is still safe - we can multiply, we can add and we stay in this smaller vector space. Such a space is going to be called a

subspace.Slide9

x

y

And how about addition?

This line is going to work, because I could add something on the line to

something else on the line and I'm still on the line. Slide10

So an example of subspace of R

2 is a line.But not just any line.

This is not a subspace !!

So which line forms a subspace in R

2

?

Line going through an origin.

Every subspace must contain zero!Slide11

What are all the possible subspaces of R

2?

all of R

2

all lines through zeroIs this line the same as R1?third subspace, that is not a whole thing, not a line, even a bit less. Which one is it?

zero vector

What are all the possible subspaces of R3?Slide12

How do subspaces come out of matrices?

How do I create some subspace from that matrix?

One subspace is from columns.

In which space are columns?

In R

3

.

OK, so I put these two columns in my subspace.

What else must be there to have a subspace?

[0 0 0] must be in subspace

All linear

combinations.

A special name for such a subspace – a

column space C(A)

.

(columns in R

3

– in that particulat case - and all their linear combinations)

What

else? Slide13

So, I have two column vectors in 3D space. What do I get, geometrically, if I take all their linear combinations?

a plane through the origin

This is simple in R

3

, but how about in R

10

? Let’s say we have a matrix 10 x 5. Five columns, each has ten components, we take their combinations. We don’t have R

5!! We have 5D hyperplane.Slide14

Main point

For matrix A, take its columns, take their combinations, all their linear combinations, and you get the column space. Slide15

Projekt I

Pá 13:00-15:00, A30data: 30.9., 7.10., 14.10., 21.10., 4.11.

21

. 1

0. – Richarda Papík (Ústav informačních studií a knihovnictví, Karlova Univerzita) na téma „Jak správně postupovat při literární rešerši“ Slide16

Project

Borromean chemistry

Genome sequencing – experimental approaches

DNA microarrays

DNA computing

Substitution matrices in sequence alignmentsRNA World TheorySlide17

Cíl projektu

Jedná o klasickou rešerši, s normálním tištěným výstupem.

Forma re

šerše

délka min 5 stran + odkazyminimálně 10 odkazů na odborné články v zahraničních anglicky psaných časopisech

je vhodné přidat i odkazy na hesla z renomovaných encyklopedií, event. učebnic - jako úvod do problematikyjazyk dokumentu - češtinaJak odevzdat rešerši

co (nutno odevzdat všechny níže uvedené položky!):1 papírový výtiskPDF verzizdrojový kód (MS Word, OpenOffice, LaTeX, ...)

jak:1 zip nebo tar.gz archiv, obsahující dokument + pdf všech odborných článků (event. kompletní HTML (tj. včetně obrázků apod.))mailem na daniel.svozil@gmail.com

termín: do 9.1. 2011 24:00, za každý další započatý týden se sníží celková výsledná známka o stupeňSlide18

prezentace

v týdnu 23.

–

27. 10.10 minut (čas bude hlídán!)v angličtiněHodnoceníHodnotí se kvalita a obsah rešerše a kvalita prezentaceKaždé se hodnotí zvlášť, výsledná známka se zkombinujeSlide19

V p

átek 15

. 10.

proběhne v

BS2 od 14:00 do cca 15:00 přednáška Richarda Papíka (Ústav informačních studií a knihovnictví, Karlova Univerzita) na téma „Jak správně postupovat při literární rešerši“ (neoficiální název)

Účast povinná !!!