MATH 117 Quiz 4

If V is a finite dimensional vector space, T an operator on V, and f(x) = a_k x^k + ... + a_1 x + a_0, how do we define
f(T)
?

The linear operator a_k T^k + a_k-1 T^k - 1 + ... + a_1 T + a_0 I_v

If V is a finite dimensional vector space, T an operator on V, and v is a vector in V, how do we define the
order ideal of v with respect to T
, Ann(T, v)?

The set of all polynomials f(x) such that v ? Ker(f(T)), that is, f(T)(v) = 0
Ann(T, v) = {f(x) ? F[x] | f(T)(v) = 0}

If V is a finite dimensional vector space, T an operator on V, and v is a vector in V, what is meant by the
minimal polynomial of T with respect to v
, �_(T, v) (x)?

The unique monic generator of Ann(T, v)

If V is a vector space, T an operator on V, and W is a subspace of V, what does it mean for W to be a
T-invariant
?

If T(w) ? W for all w ? W

If V is a vector space, T an operator on V, and W is a subspace of V, what does it mean for v to be an
eigenvector of T with eigenvalue ?
*?

T(v) = ?v

If V is a vector space, T an operator on V, and v is a vector in V, how do we define the
T-cyclic subspace of V generated by v
, <T, V>?

{f(T)(v) | f(x) ? F[x]}

If V is a vector space and T an operator on V, what is meant by the
annihilator ideal
of T?

Ann(T) consists of all polynomials f(x) such that f(T) is the zero operator
Ann(T) = {f(x) ? F[x] | f(T)(v) = 0, for all v ? V}

If V is a finite dimensional vector space and T is an operator on V, what is meant by the
minimal polynomial
of T, �_T (x)?

The unique monic polynomial of least degree in Ann(T, V)

If V is a finite dimensional vector space over the field F and T is an operator on V, what does it mean for T to be a
cyclic
?

If there is a vector v ? V such that V = <T, v>

If V is a finite dimensional vector space and T an operator on V with minimum polynomial T(x), what does it mean for a vector v in V to be a
maximal vector
for T?

A vector z such that �_(T, z) (x) = �_T (x)

If T is a linear operator on the finite dimensional vector space V and U is a T-invariant subspace of V, what is meant by a
T-complement
to U?

A T-invariant subspace W such that V = U ? W

If T is a linear operator on the finite dimensional vector space V, what does it mean when we say that T is
indecomposable
?

If no non-trivial T-invariant subspace has a T-invariant complement

If T is a linear operator on the finite dimensional vector space V, what does it mean when we say that T is
irreducible
?

If the only T-invariant subspaces are V and {0}

If T is an operator on the finite dimensional vector space V, what are the
invariant factors
of T?

The polynomials d_1 (x), d_2 (x), ..., d_r (x)

If T is an operator on the finite dimensional vector space V, what are the
elementary divisors
of T?

The polynomials p_i (x)^f_ij