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Solved ex3 from algebra hw3
2 files changed, 28 insertions, 2 deletions
| -rw-r--r-- | Year 1/Algebra/Homework3/Exercise3.pdf | bin | 29101 -> 30950 bytes | |||
| -rw-r--r-- | Year 1/Algebra/Homework3/Exercise3.tex | 30 |
diff --git a/Year 1/Algebra/Homework3/Exercise3.pdf b/Year 1/Algebra/Homework3/Exercise3.pdf Binary files differindex 64afef1..17cd19d 100644 --- a/Year 1/Algebra/Homework3/Exercise3.pdf +++ b/Year 1/Algebra/Homework3/Exercise3.pdf diff --git a/Year 1/Algebra/Homework3/Exercise3.tex b/Year 1/Algebra/Homework3/Exercise3.tex index b740b84..bd0d36d 100644 --- a/Year 1/Algebra/Homework3/Exercise3.tex +++ b/Year 1/Algebra/Homework3/Exercise3.tex @@ -53,10 +53,36 @@ \begin{align*} \psi : V \rightarrow W \quad \text{ с } \quad \psi (x_1 e_1 + x_2 e_2 + x_3 e_3) = (x_1 - 2x_2 + 3x_3)f_1 + (-2x_1 + 4x_2 - 6x_3)f_2 \end{align*} -за произволни $x_1, x_2, x_3 \in \mathbb{C}$. За кои стойности на комплексния параметър $p \in \mathbb{C}$ композицията $\psi (\varphi_1 + p\varphi_2) = \mathbb{O}$ е нулевото изображение $\mathbb{O} : V \rightarrow W$ с $\mathbb{O}(x) = \mathcal{O}_W$ за всяко $v \in V$. +за произволни $x_1, x_2, x_3 \in \mathbb{C}$. За кои стойности на комплексния параметър $p \in \mathbb{C}$ композицията $\psi (\varphi_1 + p\varphi_2) = \mathbb{O}$ е нулевото изображение $\mathbb{O} : V \rightarrow W$ с $\mathbb{O}(v) = \mathcal{O}_W$ за всяко $v \in V$. \paragraph{Решение:} -[TODO] +\begin{align*} + \psi (\varphi_1 + p\varphi_2) & = \psi ((-2x_1 + 5x_2 + x_3)e_1 + (x_1 + x_2 + x_3)e_2 + (x_1 - 2x_2 + x_3)e_3 + p(x_1 e_1 + x_3 e_2 + x_2 e_3)) = \\ + & = \psi ((-2x_1 + 5x_2 + x_3 + px_1)e_1 + (x_1 + x_2 + x_3 + px_3)e_2 + (x_1 - 2x_2 + x_3 + px_2)e_3) \\ + \text{Полагаме: } & -2x_1 + 5x_2 + x_3 + px_1 = y_1, \; x_1 + x_2 + x_3 + px_3 = y_2, \; x_1 - 2x_2 + x_3 + px_2 = y_3 \\ + & \implies \psi (\varphi_1 + p\varphi_2) = \psi (y_1 e_1 + y_2 e_2 + y_3 e_3) = (y_1 - 2y_2 + 3y_3)f_1 + (-2y_1 + 4y_2 - 6y_3)f_2 +\end{align*} + +Разглеждаме $y_1 - 2y_2 + 3y_3$: +\begin{align*} + y_1 - 2y_2 + 3y_3 & = -2x_1 + 5x_2 + x_3 + px_1 - 2(x_1 + x_2 + x_3 + px_3) + 3(x_1 - 2x_2 + x_3 + px_2) = \\ + & = -2x_1 + 5x_2 + x_3 + px_1 - 2x_1 - 2x_2 - 2x_3 - 2px_3 + 3x_1 - 6x_2 + 3x_3 + 3px_2 = \\ + & = -(1-p)x_1 - 3 (1-p)x_2 -2(1-p)x_3 = -(1-p)(x_1 + 3x_2 + 2x_3) +\end{align*} + +Разглеждаме $-2y_1 + 4y_2 - 6y_3$: +\begin{align*} + -2y_1 + 4y_2 - 6y_3 & = -2(-2x_1 + 5x_2 + x_3 + px_1) + 4(x_1 + x_2 + x_3 + px_3) - 6(x_1 - 2x_2 + x_3 + px_2) = \\ + & = 4x_1 - 10x_2 - 2x_3 - 2px_1 + 4x_1 + 4x_2 + 4x_3 + 4px_3 - 6x_1 + 12x_2 - 6x_3 - 6px_2 = \\ + & = 2(1-p)x_1 + 6(1-p)x_2 - 4(1-p)x_3 = (1-p)(2x_1 + 6x_2 - 4x_3) +\end{align*} + +Тоест, получаваме: +\begin{align*} + \psi (y_1 e_1 + y_2 e_2 + y_3 e_3) = -(1-p)(x_1 + 3x_2 + 2x_3) \cdot f_1 + (1-p)(2x_1 + 6x_2 - 4x_3) \cdot f_2 +\end{align*} + +\answerbox{\implies \text{За } p = 1, \quad \psi (v) = \mathcal{O}_W = \mathbb{O}(v) \quad \forall v \in V} \end{document}
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