Hungyi Lee Outline Reference Chapter 71 Norm amp Distance Norm Norm of vector v is the length of v Denoted Distance The distance between two vectors u and v is defined by ID: 618094
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Slide1
Orthogonality
Hung-yi LeeSlide2
Outline
Reference: Chapter 7.1Slide3
Norm & Distance
Norm
: Norm of vector v is the length of v
Denoted
Distance
: The distance between two vectors u and v is defined by
Slide4
Dot Product & Orthogonal
Dot product
: dot product of u and v is
Orthogonal
: u and v are orthogonal if
Orthogonal is actually “perpendicular”
Zero vector is orthogonal to every vector
Slide5
More about Dot Product
Let u and v be vectors, A be a matrix, and c be a scalar
if and only if
Connect norm and dot product
……
ExampleSlide6
Pythagorean
Theorem
The diagonals of a parallelogram are orthogonal.
The parallelogram is a rhombus.
Proof:
=0 if and only if u and v are orthogonal
Proof:
u
=
vSlide7
Triangle Inequality
For any vectors u and v,
Proof:
Cauchy-Schwarz Inequality