Fundamental of Linear Algebra

Definition:

• theorem ${\mathbb{R}}^n=\mathcal{N}(A)\oplus \mathcal{N}(A{)}^{\perp }=\mathcal{N}(A)\oplus \mathcal{R}(A^{⊺})$
  • direct sum of subspace and its orthogonal complement equals the whole space
• theorem for any given matrix $A\in {\mathbb{R}}^{m,n}$, it holds that $\mathcal{N}(A)\perp \mathcal{R}(A^{⊺})$ and $\mathcal{R}(A)\perp \mathcal{N}(A^{⊺})$, hence:
  • $\mathcal{N}(A)\oplus \mathcal{R}(A^{⊺})={\mathbb{R}}^n$
    • Consequently, we can decompose any vector $x\in {\mathbb{R}}^n$ as the sum of two vectors orthogonal to each other, one in range of $A^{⊺}$, and the other in the nullspace of $A$:
      • $x=A^{⊺}\xi +z,z\in \mathcal{N}(A)$
  • $\mathcal{R}(A)\oplus \mathcal{N}(A^{⊺})={\mathbb{R}}^m$
    • Similarly, we can decompose any vector $w\in {\mathbb{R}}^m$ as the sum of two vectors orthogonal to each other, one in the range of $A$, and the other in the nullspace of $A^{⊺}$:
      • $w=A\phi +\zeta ,\zeta \in \mathcal{N}(A^{⊺})$