1. Orthogonal Vector
Hung-yi Lee

2. Orthogonal Set
A set of vectors is called an orthogonal set if every pair of distinct vectors in the set is orthogonal.
Reference: Chapter 7.2
An orthogonal set?
By definition, a set with only one vector is an orthogonal set.
Is orthogonal set independent?

3. Independent?
Any orthogonal set of nonzero vectors is linearly independent.
Let S = {v1, v2, , vk} be an orthogonal set vi  0 for i = 1, 2, , k.
Assume c1, c2, , ck make c1v1 + c2v2 +  + ckvk = 0
= 0
ci = 0 ≠0
𝑐 1 = 𝑐 2 =⋯= 𝑐 𝑘 =0

4. A vector that has norm equal to 1 is called a unit vector.
Orthonormal Set
A set of vectors is called an orthonormal set if it is an orthogonal set, and the norm of all the vectors is 1
Is orthonormal set independent?
Yes −1
−4 1
A vector that has norm equal to 1 is called a unit vector.

5. Orthogonal Basis
A basis that is an orthogonal (orthonormal) set is called an orthogonal (orthonormal) basis
Orthogonal basis of R3
Orthonormal basis of R3

6. Orthogonal Projection
Orthogonal projection of a vector onto a line
𝑧∙𝑢=0 v
v: any vector
u: any nonzero vector on L
w: orthogonal projection of
v onto L , w = cu
z: v  w z L z u w
W in terms of u
Tip: 頂端；尖端
𝑣−𝑤 ∙𝑢
= 𝑣−𝑐𝑢 ∙𝑢
=𝑣∙𝑢−𝑐𝑢∙𝑢
=𝑣∙𝑢−𝑐 𝑢 2 =0
𝑐= 𝑣∙𝑢 𝑢 2
𝑤=𝑐𝑢= 𝑣∙𝑢 𝑢 2 𝑢
= 𝑣− 𝑣∙𝑢 𝑢 2 𝑢
Distance from tip of v to L :
𝑧 = 𝑣−𝑤

7. Orthogonal Projection
𝑐= 𝑣∙𝑢 𝑢 2
𝑤=𝑐𝑢= 𝑣∙𝑢 𝑢 2 𝑢
Example: L u v w z
𝑧∙𝑢=0
L is y = (1/2)x
𝑤= 𝑣∙𝑢 𝑢 2 𝑢 =
4 1 −
= −4 =
𝑣= 4 1
𝑢= 2 1

8. Orthogonal Basis 𝑐= 𝑣∙𝑢 𝑢 2 𝑤=𝑐𝑢= 𝑣∙𝑢 𝑢 2 𝑢
𝑐= 𝑣∙𝑢 𝑢 2
𝑤=𝑐𝑢= 𝑣∙𝑢 𝑢 2 𝑢
Let 𝑆= 𝑣 1 , 𝑣 2 ,⋯, 𝑣 𝑘 be an orthogonal basis for a subspace V, and let u be a vector in V.
𝑢= 𝑐 1 𝑣 1 + 𝑐 2 𝑣 2 +⋯+ 𝑐 𝑘 𝑣 𝑘
𝑢∙ 𝑣 𝑣 1 2
𝑢∙ 𝑣 𝑣 2 2
𝑢∙ 𝑣 𝑘 𝑣 𝑘 2
Proof
To find 𝑐 𝑖
𝑢∙ 𝑣 𝑖 = 𝑐 1 𝑣 1 + 𝑐 2 𝑣 2 +⋯+ 𝑐 𝑖 𝑣 𝑖 +⋯+ 𝑐 𝑘 𝑣 𝑘 ∙ 𝑣 𝑖
= 𝑐 1 𝑣 1 ∙ 𝑣 𝑖 + 𝑐 2 𝑣 2 ∙ 𝑣 𝑖 +⋯+ 𝑐 𝑖 𝑣 𝑖 ∙ 𝑣 𝑖 +⋯+ 𝑐 𝑘 𝑣 𝑘 ∙ 𝑣 𝑖
𝑐 𝑖 = 𝑢∙ 𝑣 𝑖 𝑣 𝑖 2
= 𝑐 𝑖 𝑣 𝑖 ∙ 𝑣 𝑖
= 𝑐 𝑖 𝑣 𝑖 2

9. Example
Example: S = {v1, v2, v3} is an orthogonal basis for R3

10. Orthogonal Basis
Let 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑘 be a basis of a subspace V. How to transform 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑘 into an orthogonal basis 𝑣 1 , 𝑣 2 ,⋯, 𝑣 𝑘 ?

11. Visualization