Linear Algebra › Section F · Orthogonality and least squares › Topic F.1
Inner product, length, and orthogonality
Length, distance and right angles are easy to picture in the plane, but data often lives in many more dimensions. One simple operation, the inner product, carries all three ideas into every space of vectors, and the rest of this section builds on it.
In this topic
- Why vectors need geometry
- The inner product
- Length and unit vectors
- Worked example: normalising a vector in R4
- Distance between vectors
- Orthogonal vectors
- The Pythagorean theorem in Rn
- Angles, Cauchy-Schwarz and the triangle inequality
- Worked example: the angle between two vectors
- Applied example: similarity of documents
- Common mistakes to avoid
- Key takeaways
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