Bilinear Forms

We are often interested not just in applying linear functions to vectors, but also in combining vectors from different vector spaces together in a linear way. We now introduce a way of doing exactly this.

Definition 7.3.1. Suppose V and W are vector spaces over the same field 𝕂. Then a function f : V Γ— W β†’ 𝕂 is called a bilinear form if it satisfies the following properties:

  1. It is linear in its first argument:

    1. f(v1 + v2, w) = f (v1, w) + f(v2, w) and

    2. f (cv1, w) = cf (v1, w) for all c ∈ 𝕂, v1, v2 ∈ V, and w ∈ W.

  2. It is linear in its second argument:

    1. f (v, w1 + w2) = f (v, w1) + f(v, w2) and

    2. f (v, cw1) = cf(v, w1) for all c ∈ 𝕂, v ∈ V, and w1, w2 ∈ W. β–‘

In words the definition simply says that f is a bilinear form exactly if it becomes a linear form when one of its inputs is held constant. That is, for every fixed vector w ∈ W the function gw: V β†’ 𝕂 defined by gw(v) = f (v, w) is a linear form, and similarly for every fixed vector v ∈ V the function hv : W β†’ 𝕂 defined by hv (w) = f(v, w) is a linear form.

Note: The function f (x, y) = xy is a bilinear form but not a linear transformation. Linear transformations must be linear β€œas a whole”, whereas bilinear forms just need to be linear with respect to each variable independently.

Example 7.3.2. (The Real Dot Product is a Bilinear Form). Show that the function f : ℝn Γ— ℝn β†’ ℝ defined by

f (v, w) = v β‹… w  ∀v, w ∈ ℝn

is a bilinear form.

Solution. From linearity of the dot product, it follows that the map is linear in the fist argument. From symmetry of the dot product, since v Β· w = w Β· v, the dot product is a linear form also in the second argument.  β– 

Example 7.3.3. Let V be a vector space over a field 𝕂. Show that the function g : V* Γ— V β†’ 𝕂 defined by

g(f, v) = f (v)  for all  f ∈ V*, v ∈ V

is a bilinear form.

Solution: We just notice that g is linear in each of its input arguments individually. For the first input argument, we have

g(f1 + cf2 , v) = (f1 + cf2)(v) = f1 (v) + cf2 (v) = g(f1, v) + cg(f2, v),

for all f1, f2 ∈ V*, v ∈ V, and c ∈ 𝕂 simply from the definition of addition and scalar multiplication of functions. Similarly, for the second input argument we have

g(f, v1 + cv2) = f (v1 + cv2) = f (v1) + cf(v2) = g(f, v1) + cg(f, v2),

for all f ∈ V*, v1, v2 ∈ V, and c ∈ 𝕂 since each f ∈ V* is (by definition) linear.  β– 

«The Dual Space Index Multilinear Transformations»