Question

Let F be the field of two elements and let V be a two dimensional vector space over F. How many vectors are there in V?How many one dimensional subspaces? How many different bases are there?

Answer #1

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 an n-dimensional vector space. Let W and W2 be unequal
subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and
dim(Win W2) = n - 2.

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

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)?

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 U and W be subspaces of a nite dimensional vector space V
such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈
W}.
(1) Prove that U + W is a subspace of V .
(2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases
of U and W respectively. Prove that U ∪ W...

Let U and W be subspaces of a finite dimensional vector space V
such that V=U⊕W. For any x∈V write x=u+w where u∈U and w∈W. Let
R:U→U and S:W→W be linear transformations and define T:V→V by
Tx=Ru+Sw
.
Show that detT=detRdetS
.

a)Suppose U is a nonempty subset of the vector space V over
field F. Prove that U is a subspace if and only if cv + w ∈ U for
any c ∈ F and any v, w ∈ U
b)Give an example to show that the union of two subspaces of V
is not necessarily a subspace.

Let V be a vector space and let U1, U2 be two subspaces of V .
Show that U1 ∩ U2 is a subspace of V . By giving an example, show
that U1 ∪ U2 is in general not a subspace of V .

(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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