Linear Algebra: Orthogonal Vectors
The previous article discussed Euclidean vector space, inner product, norm, distance, and angle in Rⁿ. This article will introduce orthogonal vectors, orthogonal projection, and Linear Transformation of vectors.
Orthogonal Vector
- Two vectors u and v are called orthogonal if u.v = 0
- If u and v are orthogonal vectors with Euclidean inner product, then
Orthogonal Projection
- Maps a vector onto its orthogonal projection on a line or a plane
Linear Transformation
- Functions f map Rⁿ into Rᵐ and denoted by f: Rⁿ → Rᵐ:
- If all the functions fᵢ are linear, then transformation T is called a linear transformation and these linear equations can be expressed by matrix form W = AX. Therefore, the above transformations can be written as:
Example transformation:
- Given transformation equations, w₁ = x₁ + x₂ + x₃, w₂ = x₁- x₂- x₃, the transformation will map R³ to R². T: R³ → R² (w₁, w₂) = T(x₁, x₂, x₃) = (x₁ + x₂ + x₃, x₁- x₂- x₃)
Composition of Linear Transformation
- If T₁: Rⁿ → Rᵏ and T₂: Rᵏ → Rᵐ are linear transformations, which are
Some Common Transformations on Vectors
Reflection
- Maps a vector into its symmetric image about a line or a plane
- A vector's reflection about the x-axis
Given u = (x, y)
Dilation or Contraction
- T(u) = ku is called dilation if factor k ≥ 1 and a contraction if 0 ≤ k ≤ 1. Note: k is a nonnegative scalar.
Rotation
- Rotate a vector through a fixed angle θ
Composition of Reflection and Dilation
- A reflection about the x-axis, followed by a dilation with factor 2.
Recommended Reading
Linear Algebra: Systems of Linear Equations and Matrices, with Python
Linear Algebra: Matrix Operations and their Properties, with Python
Linear Algebra: Finding Inverse Matrix, with Python
Linear Algebra: LU Decomposition, with Python
Linear Algebra: Euclidean Vector Space
Linear Algebra: General Vector Space
Linear Algebra: Discovering Eigenvalues and Eigenvectors for Diagonalization
References
[2] National Chung Cheng University Lecture – Wei-Ta Chu, Euclidean Vector Space, 2008
[3] National Chung Cheng University Lecture – Wei-Ta Chu, Angle, and Orthogonality, 2008
