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Orthogonal and orthonormal bases

Inner products & orthogonality

Japanese school year: University year 2

What you learn

Learn to define orthogonality using inner products and construct orthonormal bases consisting of mutually perpendicular unit vectors. These bases are fundamental in geometric projections and data analysis. Prior understanding of inner products, vector norms, and bases is recommended.

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Key points

Use this formula to determine whether two vectors u\boldsymbol{u} and v\boldsymbol{v} are perpendicular (orthogonal). The bracket denotes their inner product, which equals 0 when they are orthogonal.

⟨u,v⟩=0\langle \boldsymbol{u}, \boldsymbol{v} \rangle = 0

This formula calculates the length (norm ∥v∥\|\boldsymbol{v}\|) of vector v\boldsymbol{v} using the inner product. It takes the square root of the inner product of v\boldsymbol{v} with itself; a vector of length 1 is a unit vector.

∥v∥=⟨v,v⟩\|\boldsymbol{v}\| = \sqrt{\langle \boldsymbol{v}, \boldsymbol{v} \rangle}

This formula defines an orthonormal system, where all vectors ei\boldsymbol{e}_i have length 1 and are mutually perpendicular. The symbol δij\delta_{ij} is 1 when ii equals jj, and 0 when ii and jj differ.

⟨ei,ej⟩=δij\langle \boldsymbol{e}_i, \boldsymbol{e}_j \rangle = \delta_{ij}

Use this formula to decompose any vector x\boldsymbol{x} into components along orthonormal basis vectors e1\boldsymbol{e}_1 to en\boldsymbol{e}_n. Taking the inner product directly gives the coordinate along each basis vector.

x=∑i=1n⟨x,ei⟩ei\boldsymbol{x} = \sum_{i=1}^n \langle \boldsymbol{x}, \boldsymbol{e}_i \rangle \boldsymbol{e}_i

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