# Gram-Schmidt Process

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## Gram-Schmidt Process

Wanho Choi (wanochoi.com)

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Gram-Schmidt Process

Input:

Non-orthogonal set of independent vectors

Output:

Orthogonal set of vectors

It takes a linearly independent vector set for and generates an orthogonal set

that spans the same -dimensional subspace of as .

k S = {v1, ⋯, vk} k ≤ n

S′ = {u1, ⋯, uk}

k IRn S

{vi}

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Gram-Schmidt Process

v v2

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Gram-Schmidt Process

v = u

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Gram-Schmidt Process

v = u

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Gram-Schmidt Process

v1 = u1 v2

e1 = u1 u

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Gram-Schmidt Process v1 = u1 v2 e1 = u1 u e1 ⋅v2

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Gram-Schmidt Process v1 = u1 v2 e1 = u1 u e1 ⋅v2 = proju1

### ( )

v2

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Gram-Schmidt Process v1 = u1 v2 e1 = u1 u e1 ⋅v2 = proju1

### ( )

v2 u2 = v2 − proju 1

### ( )

v2

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Gram-Schmidt Process v1 = u1 v2 e1 = u1 u e1 ⋅v2 = proju1

### ( )

v2 u2 = v2 − proju 1

### ( )

v2 e2 = u2 u2

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Gram-Schmidt Process u2 = v2 − proju 1

### ( )

v2 u1 = v1 u3 = v3 − proju 1

v3 − proju2

### ( )

v3 ! uk = vkproju j

vk j=1 k

## ∑

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Gram-Schmidt Process u2 = v2 − proju 1

### ( )

v2 u1 = v1 u3 = v3 − proju 1

v3 − proju2

### ( )

v3 ! uk = vkproju j

vk j=1 k

## ∑

e1 = u1 / u1 e2 = u2 / u2 e3 = u3 / u3 ek = uk / uk !

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Gram-Schmidt Process u2 = v2 − proju 1

### ( )

v2 u1 = v1 u3 = v3 − proju 1

v3 − proju2

### ( )

v3 ! uk = vkproju j

vk j=1 k

## ∑

e1 = u1 / u1 e2 = u2 / u2 e3 = u3 / u3 ek = uk / uk !

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