If V (dimension k-1) is a subspace of W (dimension K),
and V has an orthonormal...
If V (dimension k-1) is a subspace of W (dimension K),
and V has an orthonormal basis {v1,v2.....vk-1}. Work out a
orthonormal basis of W in terms of that of V and the orthogonal
complement of V in W.
Let (V, ||·||) be a normed space, and W a
dNormV,||·|| -closed vector subspace of
V.
(a) Prove that a function |||·||| : V /W → R≥0 can be
consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| :
R≥0 | w ∈ W}).
(b) Prove that |||·||| is a norm on V /W.
(c) Prove that if (V, ||·||) is a Banach space, then so is (V
/W, |||·|||)
Let V be a vector space of dimension 1 over a field k and choose
a fixed nonzero element voe V, which is therefore a basis. Let W be
any vector space over k and let woe W be an arbitrary vector. Show
that there is a unique linear transformation T: V → W such that
T(v)= wo. [Hint: What must T(Avo) be?)
consider the subspace
W=span[(4,-2,1)^T,(2,0,3)^T,(2,-4,-7)^T]
Find
A) basis of W
B) Dimension of W
C) is vector v=[0,-2,-5]^T contained in W? if yes espress as
linear combantion
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).
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...
Let W be a discrete random variable and Pr(W = k) = 1/6, k = 1,
2 ,....., 6. Define
X =
{
W, if W <= 3;
1, if W >= 4;
}
and Y =
{
3, if W <= 3;
7 -W, if W >= 4;
}
(a) Find the joint probability mass function of (X, Y ) and compute
Pr(X +Y = 4).
(b) Find the correlation Cor(X, Y ). Are X and Y independent?
Explain.