Linear Functionals

Question 1

What is a linear function on V?

Answer 1

A map Z from V to the scalars F, there exists a unique vector v in V such that Z(u) = <u> for all u in V</u>

Question 2

What does L(V, W) represent?

Answer 2

The set of linear maps from V to W

Question 3

What is the adjoint of map T from V to W?

Answer 3

The map T* from W to V defined by:

=

Question 4

If T is linear, what can we say about its adjoint?

Answer 4

If T is linear, T* is also linear

Question 5

What is the adjoint additivity property?

Answer 5

(S + T)* = S* + T*

Question 6

What is the adjoint conjugate homogeneity property?

Answer 6

(aT)* = conjugate(a)T*

Question 7

What is the adjoint of an adjoint?

Answer 7

(T) = T

Question 8

What is the adjoint of the identity?

Answer 8

I* = I where I is the identity on V

Question 9

What is the adjoint product property?

Answer 9

(ST)* = TS

Question 10

If T is invertible, what do we know about its adjoint?

Answer 10

If T is invertible, then T* is invertible

Question 11

How can we rewrite null(T*)?

Answer 11

(range(T))^⊥

Question 12

How can we rewrite range(T*)?

Answer 12

(null(T))^⊥

Question 13

How can we rewrite null(T)?

Answer 13

(range(T*))^⊥

Question 14

How can we rewrite range(T)?

Answer 14

(null(T*))^⊥

Question 15

What is the definition of self-adjoint?

Answer 15

An operator T, equal to its adjoint. T = T*

Question 16

What do we know about the eigenvalues of a self-adjoint operator?

Answer 16

Every eigenvalue of a self-adjoint operator is real and must exist

Question 17

When is “if = 0 for all v then T = 0” valid?

Answer 17

If V is a complex inner-product space

Question 18

When is “T self-adjoint if and only if is real” valid?

Answer 18

If V is a complex inner-product space

Question 19

What can we infer if T is a self-adjoint operator and = 0?

Answer 19

T = 0

Question 20

If T is self-adjoint and a, b are real numbers such that a^2 < 4b, what can we infer?

Answer 20

T^2 + aT + bI is invertible

Question 21

What is the spectral theorem?

Answer 21

If T is self-adjoint on V, then V has an orthonormal basis consisting of eigenvectors in T.