Question

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 .

Answer #1

1. Let T be a linear transformation from vector spaces
V to W.
a. Suppose that U is a subspace of V,
and let T(U) be the set of all vectors w in W
such that T(v) = w for some v in V. Show that
T(U) is a subspace of W.
b. Suppose that dimension of U is n. Show that
the dimension of T(U) is less than or equal to
n.

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...

Construct a linear transformation T : V → W, where V and W are
vector spaces over F such that the dimension of the kernel space of
T is 666. Is such a transformation unique?
Give reasons for your answer.

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

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)...

Suppose V and W are two vector spaces. We can make the set V × W
= {(α, β)|α ∈ V,β ∈ W} into a vector space as follows:
(α1,β1)+(α2,β2)=(α1 + α2,β1 + β2)
c(α1,β1)=(cα1, cβ1)
You can assume the axioms of a vector space hold for V × W
(A) If V and W are finite dimensional, what is the dimension of
V × W? Prove your answer.
Now suppose W1 and W2 are two subspaces of V ....

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 →...

Let V and W be vector spaces and let T:V→W be a linear
transformation. We say a linear transformation S:W→V is a left
inverse of T if ST=Iv, where ?v denotes the identity transformation
on V. We say a linear transformation S:W→V is a right inverse of ?
if ??=?w, where ?w denotes the identity transformation on W.
Finally, we say a linear transformation S:W→V is an inverse of ? if
it is both a left and right inverse of...

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

(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

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