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?)
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
Let W be a discrete random variable and Pr(W = k) = 1/6, k = 1,
2 ,....., 6. Define
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?