Shared Gaussian Process Latent Variables Models - Oxford Brookes ...

2.3. LINEAR ALGEBRA 19
observed representation Y.
2.3 Linear Algebra
A linear mapping T from vector-space U to vector-space V , T : U → V is
represented in matrix form as,
T(x) = Ax. (2.5)
This means that the mapping T , represented by matrix A carries elements from
vector space U to vector space V . The image im(T) of a mapping defines the set
of all values the map can take,
im(T) = {T(x) : x ∈ U} ⊂ V. (2.6)
Similarly the kernel kern(T) is the set of all values that maps to zero,
kern(T) = {x : x ∈ U|T(x) = 0} ⊂ U. (2.7)
The kernel and the image of a linear mapping are related through the Rank-Nullity
Theorem:
dim(U) = dim(im(A)) + dim(kern(A)). (2.8)
Intuitively this means that the number of dimensions needed to correctly represent
the degrees of freedom of the data is given by subtracting the number of dimen-
sions required to represent the null-space of the representation from the number of