Linear Algebra

Question 1

Linearly independent SET

Answer 1

every list of distinct elements of S is linearly

independent

Question 2

Finite Dimensional Vector space

Answer 2

if there exists a finite set S such that Span(S)=V

also has a finite basis

Question 3

Span

Answer 3

• V ~ v-space
• S subset of V
  Span(S) is the set of all linear combinations of S

Question 4

Spanning Set

S is a spanning set for V

Answer 4

any v in V
can be expressed as
v=a1s1+...+ansn
a1,...,an scalars
s1,...,sn in S
then V=Span(S)

Question 5

rank(T)

Answer 5

dim(Im(T))

Question 6

null(T)

Answer 6

dim(Ker(T))

Question 7

Invertible matrix condition

Answer 7

det(A)=/=0

Question 8

Cayley Hamilton Theory

Answer 8

• U f.d v-space
• T:U->U
• P_T characteristic polynomial

P_T(T)=0

Question 9

T:U->U

T is diagonisable if…

Answer 9

U has a basis consisting of eigenvectors of T

dim(U)=n
U has n distinct eigenvalues

U = direct sum(E1, ...,En)
dim(U) = direct sum(dim(E1), ...,dim(En))

Question 10

injective

Answer 10

f(a1)=f(a2) -> a1=a2

Question 11

surjective

Answer 11

for all b in B, there exists an a in A:

f(a)=b

Question 12

bijective

Answer 12

injective and surjective

Question 13

V f.d v-space
U subspace of V
PROPERTIES of U

Answer 13

(i) U is finite-dimensional
(ii) dimU <= dimV
(iii) dimU = dimV if and only if U = V

Question 14

Isomorphism

Isomorphic

Answer 14

U & V subspaces
T:U->V
T bijective (T is invertible)

T~isomorphism (linear transform)
U&V are isomorphic (vector spaces)