Let V and W be Banach spaces and suppose T : V → W is a...
Let V and W be Banach spaces and suppose T : V → W is a linear
map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦
T on V is in V ∗ . Prove that T is bounded.
Questionnnnnnn
a. Let V and W be vector spaces and T : V → W a linear
transformation. If {T(v1), . . . T(vn)} is linearly independent in
W, show that {v1, . . . vn} is linearly independent in V .
b. Define similar matrices
c Let A1, A2 and A3 be n × n matrices. Show that if A1 is
similar to A2 and A2 is similar to A3, then A1 is similar to
A3.
d. Show that...
1. Let V and W be vector spaces over R.
a) Show that if T: V → W and S : V → W are both linear
transformations, then the map S + T : V → W given by (S + T)(v) =
S(v) + T(v) is also a linear transformation.
b) Show that if R: V → W is a linear transformation and λ ∈ R,
then the map λR: V → W is given by (λR)(v) =...
Let Vand W be vector spaces over F, and let B( V, W) be the set
of all bilinear forms f: V x W ~ F. Show that B( V, W) is a
subspace of the vector space of functions 31'( V x W).
Prove that the dual space B( V, W)* satisfies the definition of
tensor product, with respect to the bilinear mapping b: V x W ->
B( V, W)* defined by b(v, w)(f) =f(v, w), f E...
Prove the following:
Let V and W be vector spaces of equal (finite) dimension, and
let T: V → W be linear. Then the following are equivalent.
(a) T is one-to-one.
(b) T is onto.
(c) Rank(T) = dim(V).
Let T: V →W be a linear transformation from V to W.
a) show that if T is injective and S is a linearly independent
set of vectors in V, then T(S) is linearly independent.
b) Show that if T is surjective and S spans V,then T(S) spans
W.
Please do clear handwriting!
V and W are finite dimensional inner product spaces,T: V→W is a
linear map
1A: Give an example of a map T from R2 to itself (with the usual
inner product) such that〈Tv,v〉= 0 for every map.
1B: Suppose that V is a complex space. Show
that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉
1C: Suppose T is a linear operator on a complex space such
that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all
v).
Let f : V mapped to W be a continuous function between two
topological spaces V and W, so that (by definition) the preimage
under f of every open set in W is open in V : Y is open in W
implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that
the preimage under f of every closed set in W is closed in V . Feel
free to take V...
Let V and W be finite dimensional vector spaces over a field F
with dimF(V ) = dimF(W ) and let T : V → W be a linear map. Prove
there exists an ordered basis A for V and an ordered basis B for W
such that [T ]AB is a diagonal matrix where every entry along the
diagonal is either a 0 or a 1.
1. For a map f : V ?? W between vector spaces V and W to be a
linear map it must preserve the structure of V . What must one
verify to verify whether or not a map is linear?
2. For a map f : V ?? W between vector spaces to be an
isomorphism it must be a linear map and also have two further
properties. What are those two properties? As well as giving the
names...
Suppose V and V0 are finitely-generated vector spaces and T : V →
V0 is a linear transformation with ker(T) = {~ 0}. Is it possible
that dim(V ) > dim(V0)? If so, provide a specific example showing
this can occur. Otherwise, provide a general proof showing that we
must have dim(V ) ≤ dim(V0).