goormNLP [Least Squares]

Auspice by Goorm, Manage by DAVIAN @ KAIST

Lecture 5: Linear transformation

Transformation

• Domain (정의역): Set of all the possible values of x.
• Co-domain (공역): Set of all the possible values of y.
• Image: a mapped output y, given x.
• Range (치역): Set of all the output values mapped by each x in the domain.

=> the output mapped by a particular x is uniquely determined.

Linear Transformation

• Definition: A transformation (or mapping) T is linear if:

• standard matrix

the matrix A is called the standard matrix of the linear transformation T

ONTO and ONE-TO-ONE

1. ONTO

1. ONE-TO-ONE

example

---

Lecture 6: Least Squares

• The number u^Tv is called the inner product ** or **dot product of u and v, and it is written as:

• Properties of Inner Product:

• Vector Norm (벡터의 길이)

The length (or norm) of v is the non-negative scalar

v

defined as the square root:

• Unit vector (단위 벡터)

A vector whose length is 1 is called a unit vector.

Normalizing a vector: Given a nonzero vector v, if we divide it by its length, we obtain a unit vector as:

u is in the same direction as v, but its length is 1.

• Distance between Vectors in Rⁿ

• Inner Product and Angle Between Vectors

Inner product between u and v can be rewritten using their norms and angle:

• Orthogonal Vectors

• Back to Over-Determined System

• Least Squares: Best Approximation Criterion

• Geometric Interpretation of Least Squares.

• Normal Equation