Linear Algebra

Question 1

V is finite dimensional if and only if?

Answer 1

It contains a finite spanning set

Question 2

What does a subset W of V have to satisfy for it to be a subspace?

Answer 2

0 is in W
w,v in W implies that w+v is in W
a real, w in W implies that aw is in W

Question 3

When is f:V–>W a linear map?

Answer 3

If for all u,v in V, a complex, f(u+av) = f(u) + af(v)

Question 4

What is the kernel of f?

Answer 4

{v in V | f(v) = 0}

Question 5

What is rank(f)?

Answer 5

dim(im(f))

Question 6

What is nullity(f)?

Answer 6

dim(ker(f))

Question 7

What is the rank nullity formula for f linear and V, W finite dimensional?

Answer 7

dim(V) = rank(f) + nullity(f)

Question 8

When is λ an eigenvalue of A?

Answer 8

If there exists a non-zero poly in C^n such that Av= λv

Question 9

When is v in C^n an eigenvector of A?

Answer 9

If it is non-zero and there exists λ in C such that Av = λv

Question 10

Definition of an innner product function on V ?

Answer 10

For all u,v,w in V, k in R or C, <u> = <u> + k
For all u,v in V <u> = (real) or = conjugate (complex)
For all v in V, >=0, equality only if v=0</u></u></u>

Question 11

What is ||v||?

Answer 11

The sqrt of

Question 12

What is the characteristic polynomail, pa(x)?

Answer 12

det(xI - A)

Question 13

When is A similiar to B?

Answer 13

If there exists an invertible matrix Q st QBQ^-1 =A

Question 14

What is the Cayley Hamilton theorem?

Answer 14

Pa(A) =0

Question 15

What is the adjoint of A?

Answer 15

A* = A conjugate transpose

Question 16

What does it mean if A is self adjoint?

Answer 16

A=A*

Question 17

What does it mean if U is unitary?

Answer 17

UU* = U*U = I

Question 18

What does it mean if A is normal?

Answer 18

AA* = A*A

Question 19

Whenis the matrix A diagonisable?

Answer 19

If and only if there is a basis for C^n consisting of eigenvectors of A

Question 20

What is the dimension?

Answer 20

The number of linearly independent vectors in a basis

Question 21

What is im(f)?

Answer 21

{w in W | there exists v in V st f(v) = w}

Question 22

What is the minimal polynomial a divisor of?

Answer 22

The characteristic polynomial

Question 23

What is the minimal polynomial?

Answer 23

The monic non-zero poly of minimal degree st ma(A) = 0

Question 24

What is the rank and nullify of a matrix in REF?

Answer 24

The nullity is the number of zero rows, the rank is the number of lin ind non-zero rows

Question 25

When is a set orthonormal?

Answer 25

If = 1 when i=j, 0 if i is not equal to j

Question 26

What is the Pythagorean theorem if <u> =0</u>

Answer 26

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

Question 27

What is the Cauchy Schwartz inequality?

Answer 27

|<u>| =< ||u|| ||v||</u>

Question 28

What is the triangle inequality?

Answer 28

||u+v|| =< ||u|| + ||v||

Question 29

When is a complex matrix unitary/a real matrix orthogonal?

Answer 29

Iff it’s columns form an orthonormal basis

Question 30

A matrix is symmetric iff?

Answer 30

It is diagonalizable

Question 31

What does it mean if A is symmetric?

Answer 31

A = A transpose

Question 32

A is unitarily diagonalizable iff?

Answer 32

A is normal

Question 33

What does it mean for A to be orthogonal

Answer 33

AA^t = A^tA = I

Question 34

What is the transition matrix from A to the standard basis?

Answer 34

A

Question 35

What is the transition matrix from the standard basis to A?

Answer 35

A^-1

Question 36

What is the transition matrix from A to B?

Answer 36

(B^-1)A

Question 37

u is orthogonal to v if?

Answer 37

<u> =0</u>

Question 38

What is the geometric multiplicity nλ(A)?

Answer 38

The nullity of λI - A, ie. The number of zero rows in the REF

Question 39

What is the multiplicity mλ(A)?

Answer 39

The number of times that that λ appears in the characteristic polynomial