Linear Algebra

Question 1

What does Sylvestors criterion determine?

Answer 1

If a matrix is positive-definite

Question 2

What 3 things are equivalent in terms of if A is diagonalisable?

Answer 2

A is similar to D
The eigen values of A are distinct
A is real, symmetric

Question 3

A and B are similar if there exists an invertible matrix M such that…

Answer 3

A=M^-1BM
A and B will have the same eigenvalues

Question 4

K vectors in Rn. If k>n, k=n, k<n

Answer 4

k>n = not linearly independent
k<n = don’t span
k=n then if basis then matrix non singular

Question 5

Dim V =

Answer 5

The cardinality of any basis for V

Question 6

If u1,u2,..,uk are in v, then the span(u1,…,uk) is

Answer 6

A subspace of v

Question 7

Subspace criteria

Answer 7

Existence of origin
Closure under addition
Closure under scalar multiplication

Question 8

Properties of groups

Answer 8

Closure under .
Associativity
Identity
Inverse

Question 9

Two groups are isomorphic if

Answer 9

There exists a bijection between the groups preserving group operation

Question 10

Lagrange for groups

Answer 10

If H is a subgroup of G then |H| divides |G|

Question 11

Direct product

Answer 11

= every possible pair of elements, one from each group

Question 12

Sign of a permutation

Answer 12

+1 if sigma is the product of an even number of transpositions
-1 if an odd number

Question 13

If T is a linear map then the following are equivalent

Answer 13

ket(T)=0
T is injective
If v1,v2,..,vk are LI in v then T(v1),T(v2),…,T(vk) are LI in w

Question 14

Symmetric group sn

Answer 14

Elements are all the permutations on n distinct symbols
|Sn|= n!

Question 15

Transposition

Answer 15

A permutation that exchanges 2 elements and leaves the rest alone

Question 16

Alternating group

Answer 16

Subset of Sn containing all even permutations
|An|= n!/2

Question 17

Permutation of a set

Answer 17

A bijection to itself

Question 18

If L is hermitean then <Lv,v> is real

Answer 18

If L is anti-hermitean then <Lv,v> is purely imaginary

Question 19

Basis

Answer 19

= LI and span

Question 20

rank and nullity = k

Answer 20

rk(A) = number of leading 1’s in RREF
nullity(A) = number of columns in RREF without a leading 1

Question 21

dimU+dimV=

Answer 21

dim(U+V)+dim(U intersect W)

Question 22

If V is the direct sum of U and W then

Answer 22

If u1,uk basis of U and w1,wp basis for w then u1,uk,w1,wp basis for v
dimV=dimU+dimW
each v can be written uniquely as v=u+w

Question 23

Area of a triangle with vertices a,b,c

Answer 23

1/2 |det(b-a,c-a)|

Question 24

Scalar triple

Answer 24

a dot bxc

Question 25

If u is the orthogonal projection of v onto U then

Answer 25

||u||^2<=||v||^2

Question 26

Bessel’s inequality

Answer 26

If u1,…,unis an orthonormal basis for u and v= k1u1+k2u2+…knun then
the sum of ki^2 <=||v||^2

Question 27

If u0 is the orthogonal projection of v0 onto U then for all u

Answer 27

||u-v0||>=||u0-v0||

Question 28

Projection operator P:V to V

Answer 28

im(P) = U ker(P) = U complement thing
P^2=P
P(u)=u
(I-P)P=P(I-P)=0
P(v)= (v,u1)u1 +..+(v,uk)uk where ui is an orthonormal basis of U

Question 29

V = direct sum of U and U orthogonal complement

Question 30

U orthogonal complement =

Answer 30

{v in V: (u,v)=0 for all u in U}

Question 31

Pythagoras

Answer 31

||u+v||^2 = ||u||^2+||v||^2

Question 32

Real cauchy-schwartz inequality

Answer 32

(u,v)^2<=(u,u)(v,v)

Question 33

Complex IP

Answer 33

<u,v>= conjugate of <v,u>

Question 34

Basis =

Answer 34

maximal LI set
minimal spanning set
any u in v can be written as a sum of the vectors

Question 35

Nullspace/kernel

Answer 35

Solution space to Ax=0

Question 36

Rowspace

Answer 36

A^ty= c has a solution

Question 37

Columnspace

Answer 37

Ax=b has a solution

Question 38

Solving differential equations

Answer 38

-write as a matrix
-diagonalise A=MDM
-x= Mw
-w dot = Dw
-solve e
-use x=Mw to get back to x