Linear Algebra II (pdf, 500 kB)

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6.21. Remark. The equality of dimensions dim im(f ⊤ ) = dim im(f) is, by Prop. 6.13,
equivalent to the statement “row rank equals column rank” for matrices.
Note that [BR2] claims (in Thm. 7.8) that we also have dim ker(f ⊤ ) = dim ker(f).
However, this is false unless dim V = dim W ! (Find a counterexample!)
6.22. Interpretation in Terms of Matrices. Let us consider the vector spaces
V = F n and W = F m and a linear map f : V → W . Then f is represented by a
matrix A, and the image of f is the column space of A, i.e., the subspace of F m
spanned by the columns of A. We identify V ∗ = (F n ) ∗ and W ∗ = (F m ) ∗ with F n
and F m via the dual bases consisting of the coordinate maps. Then for x ∈ W ∗ ,
we have x ∈ (im(f)) ◦ if and only if x ⊤ y = x · y = 0 for all columns y of A, which
is the case if and only if x ⊤ A = 0. This is equivalent to A ⊤ x = 0, which says that
x ∈ ker(f ⊤ ) — remember that A ⊤ represents f ⊤ : W ∗ → V ∗ .
If we define a kernel matrix of A to be a matrix whose columns span the kernel
of A, the this says the following. Let A ∈ Mat(m × n, F ) and B ∈ Mat(m × k, F )
be matrices. Then the image of B (i.e., the column space of B) is the annihilator
of the image of A if and only if B is a kernel matrix of A ⊤ . The condition for B
to be a kernel matrix of A ⊤ means
A ⊤ B = 0 and A ⊤ x = 0 =⇒ ∃y : x = By .
Since the relation “U is the annihilator of U ′ ” is symmetric (if we identify a space
with its bidual), we find that B is a kernel matrix of A ⊤ if and only if A is a kernel
matrix of B ⊤ .
The statement (ker(f)) ◦ = im(f ⊤ ) translates into “if B is a kernel matrix of A,
then A ⊤ is a kernel matrix of B ⊤ ”, which follows from the equivalence just stated.
7. Norms on Real Vector Spaces
The following has some relevance for Analysis.
7.1. Definition. Let V be a real vector space. A norm on V is a map V → R,
usually written x ↦→ x, such that
(i) x ≥ 0 for all x ∈ V , and x = 0 if and only if x = 0;
(ii) λx = |λ|x for all λ ∈ R, x ∈ V ;
(iii) x + y ≤ x + y for all x, y ∈ V (triangle inequality).
7.2. Examples. If V = R n , then we have the following standard examples of
norms.
(1) The maximum norm:
(x1, . . . , xn)∞ = max{|x1|, . . . , |xn|} .
(2) The euclidean norm (see Section 9 below):
(x1, . . . , xn)2 = x2 1 + . . . x2 n .
(3) The sum norm (or 1-norm):
(x1, . . . , xn)1 = |x1| + · · · + |xn| .