Linear Algebra II Axioms

Question 1

Group

Answer 1

A set G, with a binary operation *,
GxG -> G, a,b -> a*b
Associativity: (a*b)*c = a*(b*c)
Identity: exists an e in G, s.t. e*a = a = a*e
Inverse: exists a b in G, s.t. a*b = e

Question 2

Field

Answer 2

A set F with two binary operations, standard addition, +, and standard multiplication, *.
F, addition is an abelian group
F \ {0}, multiplication is an abelian group
Distributivity: a.(b+c) = a.b + a.c

Question 3

Vector space

Answer 3

1st Dsitributivity law: (a+b)x = ax + ab
2nd Distributivity law: a(x+y) = ax + ay
(ab)x = a(bx)
1.x = x

Question 4

Subspace

Answer 4

0 vector exists in the subspace
Closed under addition
Closed under scalar multiplication

Question 5

Basis

Answer 5

x1, ..., xn in V forms a basis if:
* All vectors are linearly independant
* They span V

Question 6

Linear transformation

Answer 6

L(x) + L(y) = L(x+y)
L(ax) = aL(x)

Question 7

Dimension theorem

Answer 7

dim(ker) + dim(im) = dim(V)
Assuming V is finite-dimensional

Question 8

Isomorphism

Answer 8

A linear transformation, L over F, is an isomorphism to F iff L is bijective.
Two finite-dimension vector spaces are isomorphic iff they have the same dimension over the same field.

Question 9

What is meant by lambda is an eigenvalue of L

Answer 9

iff E(lambda) (L) != {0}

Question 10

Kernal criterian

Answer 10

L is injective iff ker(L) = {0}

Question 11

Cayley-Hamilton theorem

Answer 11

Let F, field, A square matrix.
Denote the characteristic polynomial of A by pA(lambda).
Then pA(A) is the zero matrix.

Question 12

What is true about x,y,z if L(x),L(y),L(z) span V

Answer 12

x,y,z are linearly independant

Question 13

State the 2 out of 3 bases criterian

Answer 13

V, a vector space over a field F, x1,…,xn in V
x1,…,xn is a basis if two out of the three hold:
* x1,…,xn are linearly independant
* x1,…,xn span V
* dim F (V) = n

Question 14

Define what a binary operation on a set S is

Answer 14

A map S x S -> S

Question 15

Defien what is meant by the matrix A representing the linear transformation L w.r.t. the bases x1,…,xn and y1,…,yn

Answer 15

L(c1.x1 + ... + cn.xn) = d1.y1 + ... + dn.yn
Where D (the matrix with row entries d(i)) = A x C (the matrix with row entries c(i))

Question 16

Define what it means to say L, a linear transformation, is diagonalisable

Answer 16

There exists a basis, x1,…,xn of V, such that the matrix A representing L with respect to the basis is diagonal.