Linear Algebra II (pdf, 500 kB)

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in a similar way as in Examples 7.2, and again they are pairwise inequivalent.
Taking f(x) = x n , we have
f1 = 1
n + 1 , f2 =
8. Bilinear Forms
1
√ 2n + 1 , f∞ = 1 .
We have already seen multilinear maps when we were discussing the determinant
in Linear Algebra I. Let us remind ourselves of the definition in the special case
when we have two arguments.
8.1. Definition. Let V1, V2 and W be F -vector spaces. A map φ : V1 × V2 → W
is bilinear if it is linear in both arguments, i.e.
∀λ, λ ′ ∈ F, x, x ′ ∈ V1, y ∈ V2 : φ(λx + λ ′ x ′ , y) = λφ(x, y) + λ ′ φ(x ′ , y) and
∀λ, λ ′ ∈ F, x ∈ V1, y, y ′ ∈ V2 : φ(x, λy + λ ′ y ′ ) = λφ(x, y) + λ ′ φ(x, y ′ ) .
When W = F is the field of scalars, φ is called a bilinear form.
If V1 = V2 = V and W = F , then φ is a bilinear form on V. It is symmetric if
φ(x, y) = φ(y, x) for all x, y ∈ V , and alternating if φ(x, x) = 0 for all x ∈ V . The
latter property implies that φ is skew-symmetric, i.e. φ(x, y) = −φ(y, x) for all
x, y ∈ V. To see this, consider
0 = φ(x + y, x + y) = φ(x, x) + φ(x, y) + φ(y, x) + φ(y, y) = φ(x, y) + φ(y, x) .
The converse holds if char(F ) = 2, since (taking x = y)
0 = φ(x, x) + φ(x, x) = 2φ(x, x) .
We denote by Bil(V, W ) the set of all bilinear forms on V × W , and by Bil(V ) the
set of all bilinear forms on V. These sets are F -vector spaces in the usual way, by
defining addition and scalar multiplication point-wise.
8.2. Examples. The standard ‘dot product’ on R n is a symmetric bilinear form
on R n .
The map that sends ( a b ) , ( c d ) ∈ R2 × R2 a c
to | b d | = ad − bc is an alternating
bilinear form on R2 .
The map (A, B) ↦→ Tr(A ⊤ B) is a symmetric bilinear form on Mat(m × n, F ).
If K : [0, 1] 2 → R is continuous, then the following defines a bilinear form on the
space of continuous real-valued functions on [0, 1]:
1
(f, g) ↦−→
0
0
1
K(x, y)f(x)g(y) dx dy .
Evaluation defines a bilinear form on V × V ∗ : (v, φ) ↦−→ φ(v).