Linear-1 9

Matrix spaces

\begin{matrix} R (A) = s p ({R_{1} (A), R_{2} (A), \dots, R_{m} (A)}) \subseteq F^{n} \\ C (A) = s p ({C_{1} (A), C_{2} (A), \dots, C_{n} (A)}) \subseteq F^{m} \\ N (A) = {x \in F^{n} | A x = 0} \subseteq F^{n} \end{matrix}

\begin{matrix} Find bases of R (A), C (A), N (A) \\ A = (\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 1 & 2 & 3 \end{matrix}) \\ A \to \dots \to (\begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) \\ ⟹ R (A) = s p ({(\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}), (\begin{matrix} 0 \\ 1 \\ 2 \end{matrix})}) \\ ⟹ C (A) = s p ({(\begin{matrix} 1 \\ 4 \\ 7 \\ 1 \end{matrix}), (\begin{matrix} 2 \\ 5 \\ 8 \\ 2 \end{matrix})}) \\ ⟹ N (A) = {x \in R^{3} | \begin{matrix} z = t \\ y = - 2 t \\ x = t \end{matrix}} = s p ({(\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix})}) \end{matrix}

\begin{matrix} Prove or disprove: \exists A \in F^{11 \times 11} : R (A) \oplus C (A) = F^{11} \\ Disproof: \\ Let A \in F^{11 \times 11} : R (A) \oplus C (A) = F^{11} \\ 11 = d i m (F^{11}) = d i m (R (A) \oplus C (A)) = \\ = \underset{r a n k (A)}{\underset{⏟}{d i m (R (A))}} + \underset{r a n k (A)}{\underset{⏟}{d i m (C (A))}} - \underset{= 0}{\underset{⏟}{d i m (R (A) \cap C (A))}} \\ ⟹ 2 r a n k (A) = 11, r a n k (A) \in N_{0} - Contradiction! \end{matrix}

\begin{matrix} A \in F^{n \times n} \\ Prove: r a n k (A) = r a n k (A^{2}) ⟺ \exists B \in F^{n \times n} : A = A^{2} B \\ Proof: \\ Let \exists B \in F^{n \times n} : A = A^{2} B \\ r a n k (A^{2}) = r a n k (A A) \leq r a n k (A) \\ r a n k (A) = r a n k (A^{2} B) \leq r a n k (A^{2}) \leq r a n k (A) \\ ⟹ r a n k (A) = r a n k (A^{2}) \\ Let r a n k (A) = r a n k (A^{2}) \\ C (A^{2}) = C (A A) \subseteq C (A) \\ d i m (C (A^{2})) = r a n k (A^{2}) = r a n k (A) = d i m (C (A)) \\ ⟹ C (A^{2}) = C (A) \\ C (A) \subseteq C (A^{2}) \\ ⟹ \forall i \in [1, n] : C_{i} (A) \in C (A^{2}) \\ C_{1} (A) = α_{1}^{1} C_{1} (A^{2}) + \dots + α_{n}^{1} C_{n} (A^{2}) \\ C_{2} (A) = α_{1}^{2} C_{1} (A^{2}) + \dots + α_{n}^{2} C_{n} (A^{2}) \\ \dots \\ C_{n} (A) = α_{1}^{n} C_{1} (A^{2}) + \dots + α_{n}^{n} C_{n} (A^{2}) \\ B = (\begin{matrix} α_{1}^{1} & \dots & α_{n}^{1} \\ ⋮ & ⋱ & ⋮ \\ α_{n}^{n} & \dots & α_{n}^{n} \end{matrix}) \\ C_{j} (A^{2} B) = \sum_{i = 1}^{n} C_{j} (A^{2}) B_{i j} = \sum_{i = 1}^{n} C_{j} (A^{2}) α_{i}^{j} = C_{j} (A) \end{matrix}