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‪Ahmed Abdelaziz‬‏

‪Ahmed A.

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Viewed Questions

Using the tabular notation shown in Figure $7.3$, specify the weather station use cases for Report status and Reconfigure. You should make reasonable assumptions about the functionality that is required here.

Software Engineering

Define the following terms: indexing field, primary key field, clustering field, secondary key freld, block anchor, dense index, and nondense (sparse) index.

Fundamentals of Database Systems

For the Sum-of-Subsets problem discussed in Chapter $5,$ can you develop an approximation algorithm that runs in polynomial time?

Foundations of Algorithms using C++ Pseudocode

Use Huffman's algorithm to construct an optimal binary prefix code for the letters in the following table. Letter : c e i r s t x Probability:0.110.220.160.120.150.100.14

Foundations of Algorithms using C++ Pseudocode

Questions asked

INSTANT ANSWER

4. If the inverse of \( \left[\begin{array}{lll}I & 0 & 0 \\ C & I & 0 \\ A & B & I\end{array}\right] \), is \( \left[\begin{array}{ccc}I & 0 & 0 \\ Z & I & 0 \\ X & Y & I\end{array}\right] \), then find \( X, Y \), and \( Z \)

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AWAITING AN EDUCATOR

5. Find an \( L U \) factorization of the following matrices (a) \( A=\left[\begin{array}{cc}2 & 5 \\ -3 & -4\end{array}\right] \) (b) \( B=\left[\begin{array}{ccc}-5 & 3 & 4 \\ 10 & -8 & -9 \\ 15 & 1 & 2\end{array}\right] \)

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AWAITING AN EDUCATOR

6. Solve the equation \( A X=b \) by using the \( L U \) factorization (a) \( A=\left[\begin{array}{ccc}3 & -7 & -2 \\ -3 & 5 & 1 \\ 6 & -4 & 0\end{array}\right], \quad b=\left[\begin{array}{c}-7 \\ 5 \\ 2\end{array}\right] \)

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INSTANT ANSWER

5. Show that \( A=\left[\begin{array}{lllll}0 & a & 0 & 0 & 0 \\ b & 0 & c & 0 & 0 \\ 0 & d & 0 & e & 0 \\ 0 & 0 & f & 0 & g \\ 0 & 0 & 0 & h & 0\end{array}\right] \) is not invertible for any values of the entries.

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INSTANT ANSWER

4. Given the matrix \( R_{\theta}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] \), with \( \theta \in \mathbb{R} \). (a) Find \( R_{\theta} R_{\theta}^{T} \) and \( R_{\theta}^{T} R_{\theta} \) hence deduce the relation between \( R_{\theta}^{T} \) and \( R_{\theta}^{-1} \). (b) Calculate \( \operatorname{det}\left(R_{\theta}\right) \). (c) If the column vector \( r=\left[\begin{array}{l}x \\ y\end{array}\right] \) represents the position of any point \( (x, y) \) in the \( x y \) plane, show that \[ r^{T} r^{\prime}=r^{T} r \] where \( r^{\prime}=\left[\begin{array}{l}x^{\prime} \\ y^{\prime}\end{array}\right]=\left(R_{\theta}\right) r \). (d) Find \( x^{\prime}, y^{\prime} \) in terms of \( x, y \), hence, using two methods, find \( x, y \) in terms of \( x^{\prime}, y^{\prime} \). (e) Given the basic unit vectors in 2D, \( e_{1}=\left[\begin{array}{l}1 \\ 0\end{array}\right] \) and \( e_{2}=\left[\begin{array}{l}0 \\ 1\end{array}\right] \), find \( e_{1}^{\prime}=\left(R_{(\pi / 4)}\right) e_{1} \) and \( e_{2}^{\prime}=\left(R_{(\pi / 4)}\right) e_{2},( \) i.e. \( \theta=\pi / 4) \). (f) Represent the vectors \( e_{1}^{\prime} \) and \( e_{2}^{\prime} \) versus \( e_{1} \) and \( e_{2} \) in the \( x y \) plane. Can you explain the geometrical effect of the matrix \( R_{(\pi / 4)} \) and in general the matrix \( R_{\theta} \), when multiplying by vector columns?

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INSTANT ANSWER

3. Find the value of \( a \) for which the following linear systems has a unique solution, an infinite number of solutions or no solution: \[ \begin{array}{c} x+y+z=2 \\ 2 x+3 y+2 z=5 \\ x+y+\left(a^{2}-1\right) z=a+1 . \end{array} \]

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INSTANT ANSWER

2. Solve for \( x, y, z \), \[ \begin{array}{r} x y-2 \sqrt{y}+3 z y=8 \\ 2 x y-3 \sqrt{y}+2 z y=7 \\ -x y+\sqrt{y}+2 z y=4 \end{array} \]

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INSTANT ANSWER

11. Show that the following set of vectors is a basis for \( M_{2 \times 2} \) \[ A_{1}=\left[\begin{array}{cc} 3 & 6 \\ 3 & -6 \end{array}\right], \quad A_{2}=\left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right], \quad A_{3}=\left[\begin{array}{cc} 0 & -8 \\ -12 & -4 \end{array}\right] ., \quad A_{4}=\left[\begin{array}{cc} 1 & 0 \\ -1 & 2 \end{array}\right] \]

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INSTANT ANSWER

10. Explain why the following sets of vectors are not bases for the indicated vector spaces (a) \( u_{1}=(1,2), u_{2}=(3,-2), u_{3}=(0,1) \) for \( \mathbb{R}^{2} \) (b) \( u_{1}=(-1,3,2), u_{2}=(6,1,1) \) for \( \mathbb{R}^{3} \) (c) \( A=\left[\begin{array}{cc}3 & 6 \\ 3 & -6\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ -1 & 0\end{array}\right], C=\left[\begin{array}{cc}0 & -8 \\ -12 & -4\end{array}\right], D=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], E=\left[\begin{array}{cc}3 & 5 \\ -1 & 8\end{array}\right] \), for \( M_{2 \times 2} \).

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AWAITING AN EDUCATOR

9. (bonus) Let \( v_{1}, v_{2}, \cdots v_{m} \) be linearly independent set of vectors in \( \mathbb{R}^{n} \) and let \( v \in \mathbb{R}^{n} \). Suppose that \( v=c_{1} v_{1}+c_{2} v_{2}+\cdots+c_{m} v_{m} \) with \( c_{i} \neq 0 \). Prove that \( \left\{v, v_{2}, \cdots v_{m}\right\} \) is linearly independent.

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