Question

Let V be a ﬁnite dimensional vector space and T ∈ L(V : V ), such that, T3 = 0. a) Show that the spectrum of T is σ(T) = {0}. b) Show that T cannot be diagonalized (unless we are in the trivial case T = O).

Answer #1

(3) Let V be a finite dimensional vector space, and let T: V® V
be a linear transformation such that rk(T) = rk(T2).
a) Show that ker(T) = ker(T2).
b) Show that 0 is the only vector that lies in
both the null space of T, and the range space of T

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 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.

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation. Fill in the six blanks
to give bounds on the sizes of the
dimension of ker(φ) and the dimension of im(φ).
3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V ) = n and dim(W) = m, and
let φ : V → W...

Suppose V is a ﬁnite dimensional inner product space. Prove that
every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V
, is an isomorphism.

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 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.

5. Let V be a finite-dimension vector space and T : V → V be
linear. Show that V = im(T) + ker(T) if and only if im(T) ∩ ker(T)
= {0}.

Let V be an n-dimensional vector space. Let W and W2 be unequal
subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and
dim(Win W2) = n - 2.

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