The image shows a page from a notebook with handwritten mathematical equations. The notebook is open, with the right page filled with calculations and the left page partially visible with more equations. The handwriting is in blue ink. The top of the page has a heading that appears to be "Module 2". Below that, there are several numbered sections (circled) with mathematical formulas and proofs. Section (i) shows a property $\phi(0) = I$, where $\phi(t) = e^{At}$ and $\phi(0) = e^{A0} = I$. Section (2) shows a property $\phi^{-1}(t) = \phi(-t)$. The proof involves $(e^{At})^{-t} = e^{-At}$, which is stated to be equal to $\phi(-t)$. Section (3) shows a property $\phi(t_1+t_2) = \phi(t_1)\phi(t_2)$. The proof begins by stating $\phi(t_1+t_2) = e^{A(t_1+t_2)}$ and then expands it to $e^{At_1} \cdot e^{At_2}$, which is equal to $\phi(t_1)\phi(t_2)$. Section (9) shows a property $(\phi(t))^n = \phi(nt)$. The proof uses $\phi(t) = e^{At}$ and then shows $(e^{At})^n = e^{nAt}$, which is equal to $\phi(nt)$. The environment is that of a desk with a notebook and some papers or books stacked in the background. The lighting appears to be coming from above. There are no people or specific city details visible in the image. in Unknown, Unknown

$The image shows a page from a notebook with handwritten mathematical equations. The notebook is open, with the right page filled with calculations and the left page partially visible with more equations. The handwriting is in blue ink. The top of the page has a heading that appears to be "Module 2". Below that, there are several numbered sections (circled) with mathematical formulas and proofs. Section (i) shows a property $\phi(0) = I$, where $\phi(t) = e^{At}$ and $\phi(0) = e^{A0} = I$. Section (2) shows a property $\phi^{-1}(t) = \phi(-t)$. The proof involves $(e^{At})^{-t} = e^{-At}$, which is stated to be equal to $\phi(-t)$. Section (3) shows a property $\phi(t_1+t_2) = \phi(t_1)\phi(t_2)$. The proof begins by stating $\phi(t_1+t_2) = e^{A(t_1+t_2)}$ and then expands it to $e^{At_1} \cdot e^{At_2}$, which is equal to $\phi(t_1)\phi(t_2)$. Section (9) shows a property $(\phi(t))^n = \phi(nt)$. The proof uses $\phi(t) = e^{At}$ and then shows $(e^{At})^n = e^{nAt}$, which is equal to $\phi(nt)$. The environment is that of a desk with a notebook and some papers or books stacked in the background. The lighting appears to be coming from above. There are no people or specific city details visible in the image.$

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Sep 20, 2026, 1:43 PM

Unknown, Unknown

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The image shows a page from a notebook with handwritten mathematical equations. The notebook is open, with the right page filled with calculations and the left page partially visible with more equations. The handwriting is in blue ink. The top of the page has a heading that appears to be "Module 2". Below that, there are several numbered sections (circled) with mathematical formulas and proofs. Section (i) shows a property $\phi(0) = I$, where $\phi(t) = e^{At}$ and $\phi(0) = e^{A0} = I$. Section (2) shows a property $\phi^{-1}(t) = \phi(-t)$. The proof involves $(e^{At})^{-t} = e^{-At}$, which is stated to be equal to $\phi(-t)$. Section (3) shows a property $\phi(t_1+t_2) = \phi(t_1)\phi(t_2)$. The proof begins by stating $\phi(t_1+t_2) = e^{A(t_1+t_2)}$ and then expands it to $e^{At_1} \cdot e^{At_2}$, which is equal to $\phi(t_1)\phi(t_2)$. Section (9) shows a property $(\phi(t))^n = \phi(nt)$. The proof uses $\phi(t) = e^{At}$ and then shows $(e^{At})^n = e^{nAt}$, which is equal to $\phi(nt)$. The environment is that of a desk with a notebook and some papers or books stacked in the background. The lighting appears to be coming from above. There are no people or specific city details visible in the image.

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