Gram Schmidt process of vector calculus linear algebra is briefly described in the slides hope you all get the required data

1. Gandhinagar Institute of Technology
Linear Algebra and Vector Calculus (2110015)
Active Learning Assignment
Semester 2
Gram–Schmidt process
Prepared By: Burhanuddin Kapadia
Guided By: Prof. Jalpa Patel
Mechanical Department (H2)

2. Outline
• Introduction
• Steps
• Examples
• Application
• Reference

3. Introduction
• It is a method for ortho-normalising a set of vectors in an inner
product space
• The process takes a finite, linearly independent set
𝑆 = {𝑣1, … , 𝑣 𝑘} for 𝑘 ≤ 𝑛
• Generates an orthogonal set 𝑆′ = {𝑢1, … , 𝑢 𝑘} that spans the same k-
dimensional subspace of 𝑹 𝑛 as 𝑆.
• This method is named after Jorgen Pedersen Gram and Erhard
Schmidt.
• In the theory of Lie group decompositions it is generalized by the
Iwasawa decomposition.

4. Introduction
• In mathematics, an orthogonal basis for an inner product space V is
a basis for V whose vectors are mutually orthogonal.
• If the vectors of anorthogonal basis are normalized, the
resulting basis is an orthonormal basis

5. Steps
• This process consists of steps that describes how to obtain an
orthonormal basis for any finite dimensional inner products.
• Let V be any nonzero finite dimensional inner product space and
suppose that {u1, u2, . . . , un} is any basis for V.
• We will form an orthogonal basis from this basis say {v1, v2, . . . , vn}

6. Steps
• Step 1: Let 𝑣1 = 𝑢1
• Step 2: Let 𝑣2 = 𝑢2 − 𝑝𝑟𝑜𝑗 𝑤1
𝑢2 = 𝑢2 −
<𝑢1,𝑣1>
𝑣11 2 𝑣2where 𝑊1 is the
space spanned by 𝑣1, and 𝑝𝑟𝑜𝑗 𝑤1
𝑢2 is the orthogonal projection of
𝑢2 on 𝑊1.
• Step 3: Let 𝑣3 = 𝑢3 − 𝑝𝑟𝑜𝑗 𝑤2
𝑢3 = 𝑢3 −
<𝑢3,𝑣1>
𝑣11 2 𝑣1 −
<𝑢3,𝑣2>
𝑣21 2 𝑣2
where 𝑊2 is the space spanned by 𝑣1 𝑎𝑛𝑑 𝑣2.
• Step 4: Let 𝑣4 = 𝑢4 − 𝑝𝑟𝑜𝑗 𝑤2
𝑢4 = 𝑢4 −
<𝑢4,𝑣1>
𝑣11 2 𝑣1 −
<𝑢4,𝑣2>
𝑣21 2 𝑣2 −
<𝑢4,𝑣3>
𝑣31 2 𝑣2 where 𝑊2 is the space spanned by 𝑣1 𝑎𝑛𝑑 𝑣2.

7. Example
• Let 𝑉 = 𝑅3 with the Euclidean inner product. We will apply the Gram-Schmidt
algorithm to orthogonalize the basis { 1, −1, 1 , 1, 0, 1 , 1, 1, 2 }
• Let 𝑢1 = 1, −1, 1 𝑢2 = 1, 0, 1 𝑢3 = (1, 1, 2)
• Following the steps:-
• Step 1: Let 𝑢1 = 𝑣1 → 𝑣1 = 1, −1, 1
• Step 2: 𝑣2 = 1, 0, 1 −
1,0,1 1,−1,1
1,−1,1 2 1, −1, 1
= 1, 0, 1 −
2
3
1, −1, 1
=
1
3
,
2
3
,
1
3
•

8. Example
• Step 3: 𝑣3 = 1, 1, 2 −
1,1,2 1,−1,1
1,−1,1 2 1, −1, 1 −
1,1,2
1
3
,
2
3
,
1
3
1
3
,
2
3
,
1
3
2
1
3
,
2
3
,
1
3
= 1, 1, 2 −
2
3
1, −1, 1 −
5
2
1
3
,
2
3
,
1
3
= −
1
2
, 0.
1
2
• Here 𝑣1 = 𝑣2 = 𝑣3 = 1, −1, 1 ,
1
3
,
2
3
,
1
3
, −
1
2
, 0.
1
2
respectively
forms an orthogonal basis for 𝑅3

9. Application
• Gram–Schmidt process to the column vectors of a full column rank
matrix yields the QR decomposition (it is decomposed into an
orthogonal and a triangular matrix).
• To obtain an orthonormal basis for an inner product space V , use the
Gram-Schmidt algorithm to construct an orthogonal basis. Then
simply normalize each vector in the basis.

10. • https://en.wikipedia.org/wiki/Gram–Schmidt_process
• http://www.math.ualberta.ca/~skalayci/Math%20102/Lecturenotes6-
8April2011.pdf