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Revision Notes
2012
unknownj
I have a kickass solution. I read my journal tons, right? Well, type up my revision notes on LJ, and revise all night. It'll work :o)

Cartesian Product is for any set A and any set B. By the Cartesian Product is meant the set:
AxB = { (a,b) | a A, b B }

Gaussian Array in terms of m linear equations and n unknowns:
a11x1 + a12x2 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
...
am1x1 + am2x2 + ... + amnxn = bm

Gaussian Array in Echelon Form
When the first number in each row that is non-zero is 1, e.g.
1 0 0 1
0 1 0 2
0 0 1 3

Linear Space V
A space with n components (x1, x2, ..., xn) which obey the operations fo addition, negation, zero and scalar multiplication.

Linear Mapping of f: v>w (where v and w are linear spaces) means a mapping fulfilling the operations of addition, negation, zero and scalar multiplication

mxn matrix A is a bracketed array written in the form:
a11 a12 ... a1n
a21 a22 ... a2n
...
am1 am2 ... amn

Basis x1, x2 ... xn V of a linear space V
Is said to form a basis of V, provided that each element (x V) may be expressed uniquely in the form
x = a1x1 + a2x2 + ... + anxn for a1, a2, ..., an R

Matrix A of a linear mapping
Phi: V>W

Isomorphism
Phi1: V>W from linear space V to a linear space W is meant a linear mapping for which there exists a linear mapping
Phi2: W>V
For which
Phi2(Phi1(x)) = x
Phi1(Phi2(x)) = x

Inner Product on a Linear Space V
Means an assignment to each ordered pair x,y V of elements of V, <x,y> R in such a way that addition, multiplication, negation and scalar multiplication hold true.

Orthoganal Complement of a subspace W of an Inner Product Space V is meant the subspace
W^(upside down T thing) = { x V | <x,y> = 0 for all y W }
(i.e. all those components x in V for which y in W are orthogonal, or something)

Change of Basis Matrix from a basis e1, e2, ..., en to basis e'1, e'2, ..., e'n of a linear space V
Is meant the matrix P of the identity linear mapping lv: v>v with respect to the basis e'1 ... e'n V in the domain, and the basis e1, ..., en V in the codomain
(what the fuck is that? I can't learn that crap, it makes no sense)

Determinant of an nxn matrix A
Is meant the R defined by:
det A = Sigma( a
(Um, likewise that, no chance, it's just a load of squiggles that don't apply to the real world of actual numbers here, unless I can find a nice example)

Adjoint adjA of an nxn matrix A
Is meant the matrix whose ijth element is the cofactor Aij of the jth element of A, since adjA is the transpose of the matrix Aij of cofactors of matrix A

Eigenvalue of a linear mapping
If A is an nxn matrix, then a non-zero vector x in Rn is called an Eigenvector of A is Ax is a scalar multiple of x. That is,
Ax = bx
For some scalar b. The scalar b is called an Eigenvalue of A and x is said to be an eigenvector corresponding to b.

Diagonalisable
Is when a linear mapping Phi:V>V from a finite dimensional linear space V to itself is said to be diagonalisable provided that there exists a basis e1,e2, ..., en V that with respect to its matrix is a diagonal matrix
b1 0 0 ... 0
0 b2 0 ... 0
0 0 b3 ... 0
0 0 0 ... bn

Characteristic Polynomial of an nxn matrix A
Is meant the |P
det(A - bI)
obtained from the matrix A of the linear mapping with respect to any basis
The |P eq.
det(A - bI) = 0
is called the characteristic eq. of the linear mapping.

Cayley Hamilton Theorem
That every matrix satisfies its characteristic matrix

Right, now I have to work out what that means....

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sorry, just need to say 'whooooooooooooooo! maths!yay!'

ahem.

Yeah, it's fun, but I don't get any of it. That's what happens when you miss 6 months of lectures :o(

heh, i understand quite a bit of that. not all of it, but most of the stuff i know ive done.

so work boy, work!

Right now I'm going through the past papers, looking at how many marks the definitions alone will get me... I'd like to avoid actual work if possible :o)

heh, that just reminds me sooo much of twat rugby maths and managment student dave who i used to be flatmates with. he did as little work as possible, and spent the majority of his time working out exactly how many marks were required to get the 40% he needs to pass. and then gets me to teach everything to him. but hah! i live not with him next year! mwahahah, hes sooooo gonna fail..

Heh, sounds worryingly like me... I do work in some stuff, but I just can't be arsed with the things I don't like. I went to all my stats stuff, all my Comp Sci stuff, my AI stuff, my Comp Soc stuff... It was just Maths Methods and Algebra and Geometry I couldn't be arsed with. But I'll scrape by - I always do...

And I thought I had a hard time in math...

(Deleted comment)
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