1. (20%) Completing the following traffic flow by using matrix multiplication. Calculate the number of cars leaving the network

$$\vec{w} = \begin{bmatrix} w1 \\ w2 \\ w3 \\ w4 \end{bmatrix} \text{ given the number of cars entering the network } \vec{x} = \begin{bmatrix} 1000 \\ 2000 \end{bmatrix}.$$

2. (20%) Let A be an n × n matrix. Use elementary row operations to transform [A l_n] to [R B], where R is a matrix in reduced row echelon form. If R = ln, in which case A is invertible, show that B = A⁻¹.(Hint: Using the fact that there exists an invertible matrix P such that PA=R)

3. (40%) Prove that: An n × n matrix A is diagonalizable if and only if there is a basis for Rⁿ consisting of eigenvectors of A. Specifically, A = PDP⁻¹ if and only if the columns of P are the basis of Rⁿ and the diagonal entries of D are eigenvalues corresponding to the respective columns of P.

4. (20%) Let A be any \( m \times n \) matrix. Prove that \( A^T A \) and A have the same null space. (Hint: Let \( \vec{v} \) be a vector in \( \mathbb{R}^n \) such that \( A^T A \vec{v} = 0 \). Observe that \( A^T A \vec{v} \cdot \vec{v} = A \vec{v} \cdot A \vec{v} = 0 \))

1. (2% for each blank, 12% in total, 本題計算過程不計分) 　Suppose two infinite planes of charge are parallel to each other (Fig. 1), one positively charged and the other negatively charged. Both planes have the same charge density σ. The direction and the magnitude of electric field at point P1 (to the left of the positively charged plane) are      (1a)      and      (1b)     , respectively. The direction and the magnitude of electric field at point P3 (to the right of the negatively charged plane) are      (1c)      and      (1d)     , respectively. The direction and the magnitude of electric field at point P2 (between the two planes) are      (1e)      and      (1f)     , respectively.

2. (2% for each blank, 6% in total, 本題計算過程不計分) A very simple electric circuit is shown in Fig. 2. Please find I₁ =      (2a)     , I₂ =      (2b)      and I₃ =      (2c)     .

(a) Faraday's Law of Induction:      (3a)     , and

(b) Lenz's Law:      (3b)     .

4. (3%) What is the origin of the magnetic field? Your explanation:      (4)     

5. (5%, 本題計算過程須計分) Find an equivalent resistor Z =      (5)      (Fig. 3 lower) for the infinite blocks of resistors (Fig. 3 upper).

6. (7%, 本題計算過程須計分) Two spherical conductors of radii r₁ and r₂ are separated by a distance much greater than the radius of either sphere. The spheres are connected by a conducting wire as shown in Fig. 4. The charges on the spheres in equilibrium are q₁ and q₂, respectively, and they are uniformly charged. Find the ratio of the magnitude of the electric field at the surface of the sphere 1 to the one of the sphere 2 is      (6a)      (3%). Please use the meaning of the ratio to explain the principle of a lightning rod (避雷計). Your explanation:      (6b)      (4%)

115年 - 115 國立臺北大學_日間學士班轉學生招生考試試題_資訊工程學系日間學士班3年級：線性代數#147399

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