Question

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.

Answer #1

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 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.

(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 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...

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 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).

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.

Let V be a finite dimensional space over R, with a positive
definite scalar product. Let P : V → V be a linear map such that P
P = P . Assume that P T P = P P T . Show that P = P T .

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...

) Let L : V → W be a linear transformation between two finite
dimensional vector spaces. Assume that dim(V) = dim(W). Prove that
the following statements are equivalent. a) L is one-to-one. b) L
is onto.
please help asap. my final is tomorrow morning. Thanks!!!!

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