Question

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}.

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 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.

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...

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 B be a (finite) basis for a vector space V. Suppose that
v is a vector in V but not in B. Prove that, if we
enlarge B by adding v to it, we get a set that
cannot possibly be a basis for V. (We have not yet formally defined
dimension, so don't use that idea in your proof.)

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

Let L : V → W be a linear transformation between two vector
spaces. Show that dim(ker(L)) + dim(Im(L)) = dim(V)

Let V be a vector space of dimension n > 0, show that
(a) Any set of n linearly independent vectors in V forms a
basis.
(b) Any set of n vectors that span V forms a basis.

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...

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