Bilinear Map - Examples

Examples

• Matrix multiplication is a bilinear map M(m,n) × M(n,p) → M(m,p).
• If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear map V × V → R.
• In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F.
• If V is a vector space with dual space V*, then the application operator, b(f, v) = f(v) is a bilinear map from V* × V to the base field.
• Let V and W be vector spaces over the same base field F. If f is a member of V* and g a member of W*, then b(v, w) = f(v)g(w) defines a bilinear map V × W → F.
• The cross product in R3 is a bilinear map R3 × R3 → R3.
• Let B : V × W → X be a bilinear map, and L : U → W be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.
• The null map, defined by B(v,w) = 0 for all (v,w) in V × W is the only map from V × W to X which is bilinear and linear at the same time. Indeed, if (v,w) ∈ V × W, then if B is linear, B(v,w) = B(v,0) + B(0,w) = 0 + 0 if B is bilinear.