Question

Consider vector spaces with scalars in the field F(could be R or C). Recall that L(V, W) is the vectors space consisting of all linear transformations from V to W.

a. Prove that L(F, W) is isomorphic to W.

b. Assume that V is a finite dimensional vectors space. Prove that L(V, F) is isomorphic to V.

c. If V is infinite dimensional, what happens to L(V, F)?

Answer #1

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 L : V → W be a linear transformation between two finite
dimensional vector spaces. Assume that dim(V) = dim(W). Prove that
the following statements are equivalent. a) L is one-to-one. b) L
is onto.
please help asap. my final is tomorrow morning. Thanks!!!!

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

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 ....

1. Assume that V is a vector space and L is a linear function V
→ V.
a. Suppose there are two vectors v and w in V such that v, w,
and v+w are all eigenvectors of L. Show that v and w share the same
eigenvalue.
b. Suppose that every vector in V is an eigenvector of L. Prove
that there is a scalar α such that L = αI.

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 U and V be subspaces of the vector space W . Recall that U ∩
V is the set of all vectors ⃗v in W that are in both of U or V ,
and that U ∪ V is the set of all vectors ⃗v in W that are in at
least one of U or V
i: Prove: U ∩V is a subspace of W.
ii: Consider the statement: “U ∪ V is a subspace of W...

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 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 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.

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