Section 15

Vectors

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v = (v_{1}, \dots, v_{n})^{T}

Fundamental object of linear algebra.

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∥ v ∥_{2} = i \sum v_{i}^{2}

Euclidean length of a vector.

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\hat{v} = \frac{v}{∥ v ∥}

Normalizes a nonzero vector.

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u \cdot v = i \sum u_{i} v_{i} = ∥ u ∥∥ v ∥ cos θ

Measures alignment and allows defining angles and orthogonality.

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cos θ = \frac{u \cdot v}{∥ u ∥∥ v ∥}

Obtains the angle between two nonzero vectors.

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d (u, v) = ∥ u - v ∥_{2}

Distance between two points/vectors.

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w = c_{1} v_{1} + \dots + c_{k} v_{k}

Builds new vectors from a given set.