Question

Check that K ( T ) is always a subspace of V .(Topological data Analysis)

Answer #1

Let V be a vector subspace of R^n for some n?N. Show that if
k>dim(V) then the set of any k vectors in V is dependent.

Let V be a subspace of Rn and let T : Rn → Rn be the
orthogonal projection onto V .
Use geometric arguments to find all eigenvectors and
eigenvalues of T . Is T diagonalisable?

Let T:V→W be a linear transformation and U be a subspace of V.
Let T(U)T(U) denote the image of U under T (i.e., T(U)={T(u⃗ ):u⃗
∈U}). Prove that T(U) is a subspace of W

Suppose that V is finite-dimensional, U ⊂ V is a subspace, and S
: U → W is a linear
map. Show that there exists a linear map T : V → W such that T u =
Su for every u ∈ U.

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 .

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}...

(a) Find k such that P(-k<T<k)=0.8 when v=7.
(b) Find k such that P(-2.069 < T < k) = 0.965 when v =
23.
(c) Find k such that P(37.652<2 <k)=0.045 when v=25.

9. Let S and T be two subspaces of some vector space V.
(b) Define S + T to be the
subset of V whose elements have the form (an
element of S) + (an element of
T). Prove that S +
T is a subspace of V.
(c) Suppose {v1, . . . ,
vi} is a basis for the
intersection S ∩ T. Extend this with
{s1, . . . ,
sj} to a basis for
S, and...

Let W be a subspace of a f.d. inner product space V and let PW
be the orthogonal projection of V onto W. Show that the
characteristic polynomial of PW is
(t-1)^dimW t^(dimv-dimw)

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

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