Lecture on Dot Product and Bilinear Forms

Introduction to the Dot Product

• Recap on the dot product: a function ( f ) that takes a vector ( w ) and returns a scalar.
• Defined for real numbers, can extend this by using transpose for any field.

Linear Functions and Forms

• The dot product function ( f ) is linear, called a linear form.
• Linearity: respects scaling and addition:
  • ( f(w_1 + w_2) = f(w_1) + f(w_2) )
  • ( f(cw) = c , f(w) )
• Can verify linearity in one step by appending a constant.
• Definition of linearity relies on properties of 0 and 1 from the field.

Alternative Definition of Linearity

• Function ( f ) can also be defined with a fixed vector ( w ) and varying ( v , (v^T w) ).
• This alternative is still linear because dot product distributes over addition and is scalar.

Bilinear Forms

• Function ( g ) from vector space cross itself to real numbers: ( g(v, w) = v \cdot w ).
• Bilinear form: linear in each argument when one is held constant.
• Definition: ( f(v, w) ) is a bilinear form if it’s linear in both arguments.

Example: Matrix Multiplication as Bilinear Form

• Bilinear form can be visualized with matrix multiplication.
• Example: For vectors from ( \mathbb{R}^2 ) and ( \mathbb{R}^3 ), using a 2x3 matrix.
  • Process: Multiply and sum elements to get a real number.
  • Align matrices properly for multiplication.

Generalizing Bilinear Forms

• Bilinear forms can be represented by a matrix.
• Any bilinear form ( f(v, w) ) can be represented as ( v^T A w ) where ( A ) is a matrix.
• Proof Outline:
  • Fix basis vectors and use linear properties to express as dot products.
  • Combine results into a matrix representation.

Important Theorem

• Any bilinear form can be rewritten in matrix form ( v^T A w ) using basis coordinates.
• Illustrates profound connection between bilinear maps and matrix operations.

Examples and Applications

• Discuss polynomial spaces and their duals as examples of bilinear forms.
• Emphasized that different vector spaces can have bilinear forms, not just ( \mathbb{R}^n ).

Conclusion

• Bilinear forms are fundamental in linear algebra, useful in various fields.
• They provide a structured way to evaluate expressions involving two vector inputs.

This lecture provided an in-depth understanding of bilinear forms and their matrix representations, showcasing the flexibility and utility of these mathematical concepts.