Understanding Orthogonality in Linear Algebra
Explore the concepts of norm, distance, dot product, and orthogonal vectors in linear algebra. Learn about the Pythagorean theorem, triangle inequality, and Cauchy-Schwarz inequality to deepen your understanding of vector operations and geometry.
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Presentation Transcript
Orthogonality Hung-yi Lee
Outline Reference: Chapter 7.1
Norm & Distance Norm: Norm of vector v is the length of v Denoted ? 2+ ?2 2+ + ?? 2 ? = ?1 Distance: The distance between two vectors u and v is defined by ? ? 12+ 22+ 32 ? = = 14 1 2 3 2 1 5 3 ? = ? = ? ? = 12+ 52+ 32 3 0 ? ? = = 35
Dot Product & Orthogonal Dot product: dot product of u and v is ? ? = ?1?1+ ?2?2+ + ???? = ??? Orthogonal: u and v are orthogonal if ? ? = 0 Orthogonal is actually perpendicular Zero vector is orthogonal to every vector
More about Dot Product Let u and v be vectors, A be a matrix, and c be a scalar ? ? = ? ? ? = 0 if and only if ? = 0 ? ? = ? ? ? ? + ? = ? ? + ? ? ? + ? ? = ? ? + ? ? ?? ? = ? ? ? = ? ?? ?? = ? ? ?? ? = ???? = ????? = ? ??? 2 Connect norm and dot product Example
Pythagorean Theorem =0 if and only if u and v are orthogonal Proof: The diagonals of a parallelogram are orthogonal. The parallelogram is a rhombus. ? + ? ? ? = 0 Proof: u = v 2 ? 2 = ?
Triangle Inequality For any vectors u and v, Proof: 2= ? 2+ 2? ? + ? 2 ? + ? 2+ 2|? ?| + ? 2 ? 2+ 2 ? ? + ? 2 Cauchy-Schwarz Inequality ? 2 ? + ?
