STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

20
interpretation of “spanning” and “independence”
in are, in terms of X,
spanning: Xc = y is solvable (in c) foranyy ∈ ;
independence: y = 0 in the above ⇒ c = 0.
If instead we begin with a matrix X, then the set
of all linear combinations of the columns of X is
a vector space (why?), called the column space
((X)), whose dimension is called the rank of
X. The independent columns of X form a basis
for (X).
Results about matrix ranks:
1) (AB) ≤ (A): Since (AB) ⊆ (A)
(why?),
(AB) =dim((AB)) ≤ dim((A)) = (A)
(The inequality follows — ass’t 1 — from Fact
2above.)
2) The rank of a matrix is at least as large as that
of any of its submatrices (you should formulate
and prove this).