Question

Let T: V -> V be a linear map such that T^{2} - 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.

Answer #1

then B=B1 U B2 is the required basis in which matrix of T is Diagonal matrix

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

(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

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, |· |v ) and (W, |· |w ) be normed vector spaces. Let T :
V → W be linear map. The kernel of T, denoted ker(T), is defined to
be the set ker(T) = {v ∈ V : T(v) = 0}. Then ker(T) is a linear
subspace of V .
Let W be a closed subspace of V with W not equal to V . Prove
that W is nowhere dense in V .

Let T : V → V be a linear operator satisfying T2 = T. Define U1
= {v ∈ V : T(v) = v} and U2 = {v ∈ V : T(v) = 0}. Prove that V = U1
⊕ U2.

（a）Suppose A ∈ M3(R) is nonzero, but A · A = 0. What are the
possibilities for the dimension of the kernel of A?
（b）Let V be a finite dimensional vector space, and U ⊂ V a
subspace.Show that there exists a linear map T:V→V with Im(T) =
U.

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 T:V-->V be a linear transformation and let T^3(x)=0 for
all x in V. Prove that R(T^2) is a subset of N(T).

1. 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.
A) If m = n and ker(φ) = (0), what is im(φ)?
B) If ker(φ) = V, what is im(φ)?
C) If φ is surjective, what is im(φ)?
D) If φ is surjective, what is dim(ker(φ))?
E) If m = n and φ is surjective, what is ker(φ)?
F)...

5. Prove or disprove the following statements.
(a) Let L : V → W be a linear mapping. If {L(~v1), . . . , L(
~vn)} is a basis for W, then {~v1, . . . , ~vn} is a basis for
V.
(b) If V and W are both n-dimensional vector spaces and L : V →
W is a linear mapping, then nullity(L) = 0.
(c) If V is an n-dimensional vector space and L : V →...

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