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Shamn S.
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Linear Algebra
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Question:

Identify $$A$$ such that $$A^4 = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$$. Consider only real elements.

Shamn S.

Note that for rotation of a vector by $$\theta$$, we transform the vector by: $$R_{\theta} = \begin{bmatrix} \cos\theta & -sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$ The identity matrix reflects either no rotation or rotation by 360$$^\circ$$. This implies that $$A^4 = R_0$$ or $$A^4 = R_{360}$$. Trivially, $$A = R_0 = \pm \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$$ is one solution. Also, note that: $$R_{\theta_1 + \theta_2} = R_{\theta_1} \cdot R_{\theta_2}$$ This implies that: $$R_{360} = R_{90} \cdot R_{90} \cdot R_{90} \cdot R_{90} = R_{90}^4 = A^4$$ $$A = \pm R_{90}$$ $$A = \pm \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$$

Java Programming
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Question:

What is a possible use case for a private static method?

Shamn S.

Consider a class method that needs to perform some computation that does not modify class state. Defining the method to be static is a good programming pattern to identify that the method is state less. If the method does computation not needed by other classes, it should be made private.

Calculus
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Question:

Compute $$\int xe^x dx$$.

Shamn S.

Consider $$\frac{d}{dx} [xe^x] = 1 \cdot e^x + x \cdot e^x$$ via product rule. In fact, the more general statement is: $$\frac{d}{dx} [xe^x + C] = 1 \cdot e^x + x \cdot e^x$$ where C is a constant. Therefore: $$xe^x + C = \int 1 \cdot e^x dx + \int x \cdot e^x dx$$ \\ Integrating both sides. $$xe^x + C - \int 1 \cdot e^x dx = \int x \cdot e^x dx$$ Finally, we arrive at: $$\int x \cdot e^x dx$$ = $$xe^x -e^x + C$$ A common technique called integration by parts leverages structuring products such that the derivate of the product yields the original integral needed. This example shows the basis for such a technique.

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