Inner Products Space - created from Mind Map

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Vector Length & Unit Vectors

Inner Products Space

Orthogonal Basis

The length, or norm, of a vector v in R^n is given by

The distance of two vectors

Dot product in R^n

Orthogonal vectors

Angle between 2 vectors

Length of a scalar multiple

Unit vector in the direction of v-bar

Satisfy 4 properties

Properties of dot product

Cauchy-Schwarz inequality

Phytagoran theorem

Triangle inequality

Satisfy 4 axioms

If V is an inner product space, then

Weighted Euclidean inner product with weights

Symmetry axiom

Additivity axiom

Homogeneity axiom

Positivity axiom

norm

If u-bar and v-bar are vectors of R^n with the Euclidean IP, then

distance

Orthogonal

An orthogonal (orthonormal) basis for a vector space is a basis that is orthogonal (orthonormal) set

Orthonormal

An orthogonal set of nonzero vectors in R^n is linearly independent

Let S={vectors u}, an orthonormal basis for R^n and v any vector in R&n

Gram-Schmidt Theorem

S3 :

S1 : Let

S4 :

S2 :