\(A^{-1}\) 을 한 번 구해두면 \(x = A^{-1}b\) 를 통해 해를 구하면 되지 않을까?

성질로써의 분류

Numpy 편의기능 및 기본조작

오늘의 문제

프로그래머스 기능개발

오늘의 문제

프로그래머스 멀쩡한 사각형

📖 Theorem 1.

Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix

📖 Theorem 10.

Let \(T : \mathbb{R^n} \to \mathbb{R^m}\) be a linear transformation. Then there exists a unique matrix A such that
\(T(\mathbf{x}) = A\mathbf{x} \mbox{ for all }\mathbf{x} \mbox{ in }\mathbb{R^n}\)
In fact, \(A\) is the \(m \times n\) matrix whose j-th column is the vector \(T(e_j)\), where \(e_j\) is the j-th column of the identity matrix in \(\mathbb{R^n}\):
\(A = \left[ T(e_1) \cdots T(e_n) \right]\)
Standard matrix for the linear transformation T

Transformation

A transformation (or function or mapping) \(T\) from \(\mathbb{R^n} \mbox{ to } \mathbb{R^m}\)
Is a rule that assignms to each vector x in \(\mathbb{R^n}\) a vector \(T(\mathbf{x})\) in \(\mathbb{R^m}\)
Domain to CoDomain → \(T : \mathbb{R^n} \to \mathbb{R^m}\)

Linearly Independent

A set of vectors \({\mathbf{v_1}, \cdots, \mathbf{v_p}}\) in \(\mathbb{R^n}\) is said to be linearly independent, if the vector equation
\(x\mathbf{v_1} + \cdots + x_p\mathbf{v_p} = 0\)
Has only the trivial solution.
trivial solution 만 있을 때, vector 앞에 있는 coefficient 가 다 0인 솔루션

• 어떤 vector set 에 LD 를 따지는 것은, \(A\mathbf{x} = 0\)을 푸는 것과 동일함
• free variable이 한 개라도 존재하면, Nontrivial solution → 해가 무수히 존재

Homogeneous Linear Systems

Trivial solution
\(A\mathbf{x} = 0\) (→ 제로 백터 ) always has at least one solution \(\mathbf{x} = 0\)