Linear Algebra 2

Question 1

what is the difference between a basis and a spanning set

Answer 1

for a basis the elements need to be linearly independent

Question 2

what is the spectrum of a linear operator T

Answer 2

the set of eigenvalues of T

Question 3

what do we denote it with

Answer 3

σ(T)

Question 4

Suppose that B is similar to A, what 3 implications can be made

Answer 4

(1) A and B have the same characteristic polynomial, pA(x) = pB(x);
(2) the algebraic multiplicity of an eigenvalue λ of A is the same as its algebraic multiplicity as an eigenvalue of B;
(3) the geometric multiplicity of an eigenvalue λ of A is xthe same as its geometric
multiplicity as an eigenvalue of B.

Question 5

what is always true about the multiplicities of any eigenvalue

Answer 5

the geometric multiplicity is less than or equal to the algebraic multiplicity: nλ ≤ mλ .

Question 6

what is the cyclic property of traces

Answer 6

trace(XYZ)=trace(YZX)=trace(ZXY)

Question 7

give me 4 properties of ajoints

Answer 7

(1) (T + U)* = T* + U*

(2) (cT)* = (conjugate of c) T*

(3) (T U)* = U* T*

(4) ( T* ) * = T.

Question 8

yh give me 4 more to do with kernel and image

Answer 8

(1) Ker T* = (Im T)⊥

(2)Ker T = (Im T*)⊥

(3) Im T* = (Ker T)⊥

(4) Im T = (Ker T*)⊥

Question 9

what is an isometry

Answer 9

U : V → W is called an isometry (or norm preserving) |Ux|w = |x|v ∀ x ∈ V.

Question 10

give me a fact about the eigenvalues of unitary operators

Answer 10

Eigenvalues of unitary operators have absolute value 1

Question 11

<Ux, Uy>w = < x, y >v ∀ x, y ∈ V.
what does this mean

Answer 11

U : V → W preserves norms if and only if it preserves inner products

Question 12

suppose, U : V → V is a unitary operator: if
{x1, . . . , xr} ⊂ V is an orthonormal set, what is another orthonormal set?

Answer 12

{Ux1, . . . , Uxr} is an orthonormal set.

Question 13

what is a unitary matrix

Answer 13

if (U*) U = I

Question 14

A matrix O ∈ Mn (R) is called orthogonal if …

Answer 14

if its transpose is equal to its inverse

Question 15

show your additivity in the first slot

Answer 15

⟨𝑢 + 𝑣, 𝑤⟩ = ⟨𝑢, 𝑤⟩ + ⟨𝑣, 𝑤⟩ for all 𝑢, 𝑣, 𝑤 ∈ 𝑉.

Question 16

what is a good relation to know with matrices and inverses to do with similarity and characteristic polynomials

Answer 16

p(R T R^−1 ) = R p(T) R^−1

Question 17

what is a monic polynomial

Answer 17

if its highest order is 1

Question 18

what is a minimal polynomial

Answer 18

a unique lowest degree monic polynomial mA(x) over C, such that mA(A) = 0

Question 19

what are 3 properties of a minimal polynomial

Answer 19

1. If q(x) is a polynomial such that q(A) = 0, then mA(x) divides q(x).
2. The roots of mA(x) = 0 are precisely the eigenvalues λ of A.
3. If A and B are similar, then mA(x) = mB(x).

Question 20

tell me all the possible forms of jordan block for a 3x3 matrix

Answer 20

do it on i pad and verify against lecture notes

Question 20

state the spectral mapping theorem

Answer 20

σ (p(A)) = p(σ(A)),

Question 20

what is a theorem about the Jordan Canonical Form

Answer 20

Every square matrix A is similar over C to a partitioned matrix of the form
| T₁ . . . 0 |
| 0 T₂ |
| 0 0 . |
| 0 0 . .Tₙ |

where Ti is a square matrix of some dimension ri × ri

T₁ 0 . . . 0 |

Question 21

For a Jordan normal form matrix T (with notation as in Theorem 1.12.1) the eigenvalues and multiplicities can be read off directly as follows cmon give me two key things to know

Answer 21

1. The eigenvalues of T are given by the diagonal entries of T, and the algebraic multiplicity mλ of an eigenvalue λ is the number of times the eigenvalue appears in the diagonal of T .
2. Given an eigenvalue λ of T its geometric multiplicity nλ is equal to the number of Jordan blocks Ti = Jλi,ri
   for which λi = λ. This will follow from the proof of
   the Jordan Canonical Form Theorem, where each Jordan block with eigenvalue λ
   will be seen to contribute one dimension to the eigenspace with eigenvalue λ.

Question 22

what is the another way of calling a matrix self-adjoint

Answer 22

Hermitian

Question 23

what can we imply if a linear operator is self - adjoint

Answer 23

- the eigenvalues are real - there exists an orthonormal basis for V consisting of eigenvectors

Question 24

what is true about linear operators over ℂ in terms of basis

Answer 24

- there exists an orthonormal basis such that the matrix T is upper triangular

Question 25

in what scenarios are upper triangular matrices normal

Answer 25

- When they are diagonal

Question 26

what is the parallelogram identity

Answer 26

|z₁ + z₂|^2 + |z₁ - z₂|^2 = 2 ||z₁| + |z₂||^2

Question 27

True or False: If A is self-adjoint, then A is not unitarily diagonalisable.

Answer 27

False

Question 28

Which statement is True 1) If T is normal, then it is unitarily diagonalisable. 2) If A is unitarily diagonalisable, then A is normal.

Answer 28

Both

Question 29

If T is normal, there is an orthonormal basis of V consisting of eigenvectors of T is true if V is over C or R?

Answer 29

over C

Question 30

What is required for us to call a self-adjoint linear operator T : V → V a positive definite

Answer 30

> 0

Question 31

what is the condition for positive semi - definite

Answer 31

≥ 0

Question 32

when is T > 0 and ≥ 0

Answer 32

- if all eigenvalues are positive - if all eigenvalues are non-negative

Question 33

how do u find the nullity of a matrix

Answer 33

- put matrix in reduced echelon form and number of non-zero rows is the dimension of the kernel

Question 34

what is the determinant of an orthogonal matrix

Answer 34

+- 1

Question 35

Any orthogonal set of non-zero vectors is linearly independent

Answer 35

True

Question 36

let V be an inner product space let S be an orthogonal set: S = {x1, ...., xn} state the generalised Pythagorean identity

Answer 36

verify against lecture notes

Question 37

Let E be the subspace of V with basis {x1, x2, ... , xn} what is the formula for the orthogonal projection of x

Answer 37

verify against lecture notes

Question 38

is the orthogonal projection a linear transformation

Answer 38

Yes

Question 39

what is another way of calling a unitary operator

Answer 39

a norm preserving linear operatorw

Question 40

what are the forms of 1x1 matrices for unitary and orthogonal ones

Answer 40

unitary: e^iθ orthogonal: +- 1

Question 41

what is the determinant of a unitary matrix

Answer 41

e^iθ and its determinant is +- 1

Question 42

what is the condition for a matrix to be unitarily diagaonalized (unitary equivalent to a diagonal matrix)

Answer 42

it has n orthonormal eigenvectors

Question 43

give me two theorems for hermitian matrices (for C and R)

Answer 43

1. Let A ∈ Mn(C) be Hermitian. Then there exists a unitary matrix U ∈ Mn(C) such that U−1AU is diagonal. 2. Let A ∈ Mn(R) be Hermitian (i.e. symmetric). Then there exists an orthogonal matrix O ∈ Mn(C) such that O−1AO is diagonal.

Question 45

true or false: For every normal matrix A ∈ Mn(C) there is a unitary matrix U ∈ Mn(C) such that U−1AU is diagonal.

Answer 45

true

Question 46

Let V be an inner product space and T : V → V a linear operator on V . Then T is normal if and only if

Answer 46

|Tx| = |T∗x| ∀ x ∈ V.