Question

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.

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 U and W be subspaces of a finite dimensional vector space V
such that V=U⊕W. For any x∈V write x=u+w where u∈U and w∈W. Let
R:U→U and S:W→W be linear transformations and define T:V→V by
Tx=Ru+Sw
.
Show that detT=detRdetS
.

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 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 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 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.

Let F be the field of two elements and let V be a two
dimensional vector space over F. How many vectors are there in
V?How many one dimensional subspaces? How many different bases are
there?

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 to W be a linear transdormation of vector
space V and W and let B=(v1,v2,...,vn) be a basis for V. Show that
if (Tv1,Tv2,...,Tvn) is linearly independent, thenT is
injecfive.

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}.

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