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....

nozzaahem.

unknownj## Re:

nozzaso work boy, work!

unknownj## Re:

nozzaunknownjgcpunk(Deleted comment)