Question

(3) Let V be a finite dimensional vector space, and let T: V® V
be a linear transformation such that rk(T) = rk(T^{2}).

a) Show that ker(T) = ker(T^{2}).

b) Show that **0** is the only vector that lies in
both the null space of T, and the range space of T

Answer #1

If any doubt pls comment

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

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 and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

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).

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 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 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.

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 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 a linear map such that T2 - I = 0
where I is the identity map on V.
a) Prove that Im(T-I) is a subset of Ker(T+I)
and Im(T+I) is a subset of Ker(T-I).
b) Prove that V is the direct sum of Ker(T-I) and
Ker(T+I).
c) Suppose that V is finite dimensional. True or false there
exists a basis B of V such that [T]B
is a diagonal matrix. Justify your answer.

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