Homogeneous Linear Systems
Trivial solution
\(A\mathbf{x} = 0\) (→ 제로 백터 ) always has at least one solution \(\mathbf{x} = 0\)
NonTrivial solution
If and only if the equation has at least one free variable → 유일하거나 무한히 많을 때
📖 Theorem2. Existence and Uniqueness Theorem
If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable
\(A\mathbf{x} = 0\) the solution set can always be expressed as
\(Span\{\mathbf{v_1}, \mathbf{v_2}, \cdots, \mathbf{v_p}\}\)
Trivial Solution : \(Span\{0\}\) → v를 0벡터로 표현
NonHomogeneous Linear Systems
\(A\mathbf{x} = \mathbf{b}\) →b is nonzero vector
📖 Theorem6.
Suppose
\(A\mathbf{x} = \mathbf{b}\) is consistent (→ 해가 최소 1개 이상일 때) And let \(\mathbf{p}\) be a solution ( → p 벡터가 솔루션이라 가정하며 )
Then the solution set of \(A\mathbf{x} = \mathbf{b}\) is the set of all vectors of the form
\(\mathbf{w} = \mathbf{p} + \mathbf{v_h}\) where \(/mathbf{v_h}\) is any solution of the homogeneous equation \(A\mathbf{x} = \mathbf{0}\)