Question

1.
Suppose that ? is a finite dimensional vector space over R. Show
that if ???(? ) is odd, then every ? ∈ L(? ) has an eigenvalue.
(Hint: use induction).

(please provide a detailed proof)

2. Suppose that ? is a finite dimensional vector space over R
and ? ∈ L(? ) has no eigenvalues. Prove that every ? -invariant
subspace of ? has even dimension.

Answer #1

Prove that if ? is a finite dimensional vector space over
C, then every ? ∈ ℒ(? ) has an eigenvalue.
(note: haven't learned the Fundamental Theorem of Algebra
yet)

Let V be a finite-dimensional vector space over C and T in L(V)
be an invertible operator in V. Suppose also that T=SR is the polar
decomposition of T where S is the correspondIng isometry and
R=(T*T)^1/2 is the unique positive square root of T*T. Prove that R
is an invertible operator that committees with T, that is
TR-RT.

Give an example with a proof of an infinite-dimensional vector
space over R

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

Suppose that V is a finite dimensional inner product space over
C and dim V = n, let T be a normal linear transformation of V
If S is a linear transformation of V and T has n distinc
eigenvalues such that ST=TS. Prove S is normal.

Let V be a finite-dimensional vector space and let T be a linear
map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)).
Prove that the range and null space of T have only the zero vector
in common

Let B be a (finite) basis for a vector space V. Suppose that
v is a vector in V but not in B. Prove that, if we
enlarge B by adding v to it, we get a set that
cannot possibly be a basis for V. (We have not yet formally defined
dimension, so don't use that idea in your proof.)

1.
Let ??(?) = ∑︀??=1 ??? denote the operation of taking the trace of
? ∈ ??(F). Show that (by constructing a proof) ??(??+??) =
???(?)+???(?) for any ?,? ∈ ??(F) and ?,? ∈ F. Conclude that ?? is
a linear transformation from ??(F) to F.
2. For a fixed ? ∈ R, determine the dimension of the subspace
of P?(R) defined by {? ∈ P?(R) | ?(?) = 0}.
3. Let ? be a finite dimensional vector space. Suppose...

Let T:V→V be an endomorphism of a finite dimensional vector
space over the field Z/pZ with p elements, satisfying the equation
Tp=T. Show that T is diagonalisable.

Let V be a finite dimensional vector space over R with an inner
product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
(c) (2 points) Recall that the norm of a vector x ∈ V...

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