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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
Lecture 5, Lecture 6
Transposes
How to write transposition 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:
1 2 4
R

3
3
1


Why is RTR symmetric?
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.
• 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: R2 – 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 plane
• 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.
• Is line given as [3 2 0] in R2 or in R3?
• So what’s in Rn?
– all vectors with n components
• We 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).
y
x
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.
Vector subspaces
• Rn is a good space, but raher large. We’ll
be interested in spaces that are inside Rn.
• 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.
y
x
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.
• So an example of subspace of R2 is a line.
• But not just any line.
This is not a subspace !!
So which line forms a subspace in R2?
Line going through an origin.
Every subspace must contain zero!
•
What are all the possible subspaces of
R 2?
1. all of R2
2. all lines through zero
– Is this line the same as R1?
3. 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
R 3?
• How do subspaces come out of matrices?
1 3
A  2 3
4 1
How do I create some subspace from that matrix?
One subspace is from columns.
In which space are columns?
In R3.
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
What else?
All linear combinations.
A special name for such a subspace – a column space C(A).
(columns in R3 – in that particulat case - and all their linear
combinations)
1 3
A  2 3
4 1
• 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 R3, but how about in
R10? Let’s say we have a matrix 10 x 5.
Five columns, each has ten components,
we take their combinations. We don’t
have R5!! We have 5D hyperplane.
Main point
• For matrix A, take its columns, take their
combinations, all their linear combinations,
and you get the column space.
Projekt I
• Pá 13:00-15:00, A30
• data: 30.9., 7.10., 14.10., 21.10., 4.11.
• 21. 10. – Richarda Papík (Ústav
informačních studií a knihovnictví, Karlova
Univerzita) na téma „Jak správně
postupovat při literární rešerši“
Project
1. Borromean chemistry
2. Genome sequencing – experimental
approaches
3. DNA microarrays
4. DNA computing
5. Substitution matrices in sequence alignments
6. RNA World Theory
Cíl projektu
– Jedná o klasickou rešerši, s normálním tištěným výstupem.
Forma rešerše
– délka min 5 stran + odkazy
– minimálně 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 problematiky
– jazyk dokumentu - čeština
Jak odevzdat rešerši
– co (nutno odevzdat všechny níže uvedené položky!):
• 1 papírový výtisk
• PDF verzi
• zdrojový 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 [email protected]
– termín: do 9.1. 2011 24:00, za každý další započatý týden se
sníží celková výsledná známka o stupeň
• prezentace
– v týdnu 23. – 27. 10.
– 10 minut (čas bude hlídán!)
– v angličtině
Hodnocení
– Hodnotí se kvalita a obsah rešerše a kvalita
prezentace
– Každé se hodnotí zvlášť, výsledná známka se
zkombinuje
• V pátek 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 !!!