Lecture on Dot Products and Linear Algebra

Introduction to Dot Products

• Traditionally introduced early in linear algebra courses.
• Deferred in this lecture series for fuller understanding under linear transformations.

Standard Introduction to Dot Products

• Numerically: Dot product of two vectors involves:
  • Pairing up coordinates.
  • Multiplying pairs together.
  • Adding the result.
• Examples:
  • 1, 2 dotted with 3, 4 => 1*3 + 2*4
  • 6, 2, 8, 3 dotted with 1, 8, 5, 3 => 6*1 + 2*8 + 8*5 + 3*3

Geometric Interpretation of Dot Products

• Imagine projecting w onto the line through the origin and tip of v.
• Multiply length of this projection by length of v to get v . w.
• Dot product interpretations:
  • Same direction => Positive
  • Perpendicular => Zero
  • Opposite direction => Negative
• Order doesn't matter: v . w = w . v due to symmetrical projection properties.

Effects of Scaling

• Scaling v by a constant affects dot product proportionally.
• Example: Scaling v by 2 doubles v . w because projection length stays the same, but v length doubles.

Numerical and Geometric Connection

• Numerical process matches coordinates, multiplies pairs, sums them.
• Deeper understanding involves duality and linear transformations from multiple dimensions to one.
• Linear transformations in 2D vector space to number line (1D output).

Linear Transformations

• Defined with a 1x2 matrix representing vectors.
• Applying transformation to a vector involves matrix vector multiplication.
• Example transformation: i-hat to 1 and j-hat to -2 transforms vector 4,3 to -2.
• Association between 1x2 matrices and 2D vectors.

Duality in Linear Algebra

• Projection of 2D vectors onto a diagonal line (represented as number line in 2D space).
• u-hat: 2D unit vector on the diagonal line, projection transformation defined by u-hat.
• i-hat and j-hat projection onto the diagonal line has symmetry matching their x and y coordinates respectively.
• 1x2 matrix formed by coordinates of u-hat for projection transformation.
• Projection transformation computationally identical to dot product with u-hat.

Dot Products with Non-Unit Vectors

• Scaling unit vectors and associated matrix scales projection proportionally.
• Dot product with non-unit vector interpreted as projection and scaling.

Significance and Summary

• Linear transformations from 2D space to number line described by dot products.
• Duality between vectors and transformations: vector can represent transformation and vice versa.
• Dot product as a geometric tool for projections and vector alignment.
• Deeper level: Translating vectors into linear transformations.

Conclusion

• Dot product translates vectors to linear transformations.
• Understanding vectors as transformations provides deeper mathematical insights.
• Upcoming topic: Cross product and duality.