Dot product & duality | Math 32A Lec2 topic

Notes on 2Blue1Brown video

Dot product defined for two vectors of the same dimension as follows:
- Pair up the coordinates of the vectors.
- Multiply the pairs together and add the results.
Numerical example of dot products:
- For vectors (1, 2) and (3, 4):
- Calculation: $1 imes 3 + 2 imes 4 = 3 + 8 = 11$
- For vectors (6, 2, 8, 3) and (1, 8, 5, 3):
- Calculation: $6 imes 1 + 2 imes 8 + 8 imes 5 + 3 imes 3 = 6 + 16 + 40 + 9 = 71$

The dot product has a geometric interpretation involving projection:
- Consider two vectors, v and w.
- Project vector w onto the line of vector v:
- The length of this projection multiplied by the length of v gives the dot product: $ext{dot product} = ext{length of projection} imes ext{length of } v$
Significance of the dot product:
- Positive: When vectors point in the same direction.
- Zero: When vectors are perpendicular (one projects to zero).
- Negative: When vectors point in opposite directions.

Order of the vectors in dot product does not affect the result:
- Projecting w onto v or v onto w yields the same numerical output.
Intuition behind this:
- If both vectors are the same length, projections are symmetrical.
- If lengths differ, scaling one vector (e.g., scaling v by factor of 2) impacts the dot product, still resulting in the same doubling effect:
- $2v \bullet w = 2(v \bullet w)$

Explains confusion on numerical process:
- The operations (multiplying coordinates and summing) relate to the concept of projection.
Introduction to the concept of duality:
- Need to explore linear transformations from multidimensional spaces to one-dimensional spaces (number lines).

Definition of linear transformations:
- Functions that take a 2D vector and output a single number.
Properties of linear transformations:
- They maintain equal spacing of points (dots).
Relation to basis vectors:
- Each transformation is determined by where unit basis vectors (i-hat and j-hat) map in the output space.
Example transformation:
- Transforming i-hat to 1 and j-hat to -2.
- For vector (4, 3):
- Obtained by the transformation:
  - Calculation: $4 imes 1 + 3 imes (-2) = 4 - 6 = -2$
Matrix Representation:
- Transformation can be represented as a 1x2 matrix.
- Tipping a 2D vector to create a 1x2 matrix reveals close relation to dot products.

Linear transformations from 2D to numbers have unique corresponding vectors:
- Transformation can also be expressed as a dot product with associated vectors.
Creation of a diagonal number line in space:
- Projecting 2D vectors onto this embedded line.
Maintains linear structure (evenly spaced dots).
Outputs are scalar values, confirming that the process defines a linear transformation.
Projection symmetries provide insight into relationships between vectors:
- Upon projection, properties of i-hat and j-hat reveal echoes of the original vector u-hat's coordinates.

Entries of the 1x2 matrix correspond to coordinates of u-hat.
Projection transformations, analogous to dot products, reinforce the connection between linear transformations and related vectors.

When scaling a unit vector (e.g., u-hat scaled by 3), the resulting transformation affects projections of any vector:
- Outputs are scaled versions of the previous results.
Example of transformations with a scaled vector:
- Transformation is now taking the projection and adjusting its length according to the scaling of the vector.

In summary:
- Dot products relate to vector projections geometrically.
- They also encode transformations, allowing for deeper understanding of vectors as linear transformation embodiments.
- The notion of duality exemplifies a natural correspondence in mathematics, showcasing interwoven relationships between vectors and transformations.