Linear Functionals Flashcards
What is a linear function on V?
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>
What does L(V, W) represent?
The set of linear maps from V to W
What is the adjoint of map T from V to W?
The map T* from W to V defined by:
=
If T is linear, what can we say about its adjoint?
If T is linear, T* is also linear
What is the adjoint additivity property?
(S + T)* = S* + T*
What is the adjoint conjugate homogeneity property?
(aT)* = conjugate(a)T*
What is the adjoint of an adjoint?
(T) = T
What is the adjoint of the identity?
I* = I where I is the identity on V
What is the adjoint product property?
(ST)* = TS
If T is invertible, what do we know about its adjoint?
If T is invertible, then T* is invertible
How can we rewrite null(T*)?
(range(T))^⊥
How can we rewrite range(T*)?
(null(T))^⊥
How can we rewrite null(T)?
(range(T*))^⊥
How can we rewrite range(T)?
(null(T*))^⊥
What is the definition of self-adjoint?
An operator T, equal to its adjoint. T = T*
What do we know about the eigenvalues of a self-adjoint operator?
Every eigenvalue of a self-adjoint operator is real and must exist
When is “if = 0 for all v then T = 0” valid?
If V is a complex inner-product space
When is “T self-adjoint if and only if is real” valid?
If V is a complex inner-product space
What can we infer if T is a self-adjoint operator and = 0?
T = 0
If T is self-adjoint and a, b are real numbers such that a^2 < 4b, what can we infer?
T^2 + aT + bI is invertible
What is the spectral theorem?
If T is self-adjoint on V, then V has an orthonormal basis consisting of eigenvectors in T.
