(b) (6 pts) Describe the eigenvectors corresponding to each eigenvalue of \( T \).

(c) (8 pts) Find an ordered basis \( \beta \) for \( \mathsf{M}{n\times n}(\mathbb{R}) \) such that \( [T]{\beta} \) is a diagonal matrix.

5. (15 pts) Let$$A = \begin{pmatrix} 3 & 1 & 2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}.$$Find a Jordan canonical form $J$ of $A$ and an invertible matrix $Q$ such that $J = Q^{-1}AQ$.

6. (15 pts) State and prove the Cayley-Hamilton Theorem.

7. (15 pts) Let \( T \) be a projection on a finite-dimensional inner product space \( V \) such that \( \| T(x) \| \leq \| x \| \) for all \( x \in V \). Prove that \( T \) is an orthogonal projection.

6 (15) Let \( M \) be a compact metric space and \( \Phi : M \to M \) be such that \( d(\Phi(x), \Phi(y)) < d(x, y) \) for all \( x, y \in M \), \( x \neq y \). Show that \( \Phi \) has a unique fixed point.

1. (15%) For the circuit shown in Fig. 1 first, evaluate T i(s)=Vi(s)/Vs(s) and the correspondingcutoff (corner) frequency. Second, evaluate T o(s)=Vo(s)/Vi(s) and the corresponding cutofffrequency. Put each of the transfer functions in the standard form and combine them to form theoverall transfer function, T(s)=Ti(s)×To(s). Provide a Bode magnitude plot for ∣T(jw)∣. Whatis the bandwidth between 3-dB cutoff points?

(a) Find the resistances looking into node 1, R1; node 2, R2; node 3, R3; and node 4, R4.

(b) Find the current I1, I2, I3, and I4 in terms of the input current I.

3. (15%) Fig. 3 shows a circuit that performs a low-pass STC function. Such a circuit is known asa first-order low-pass active filt er. Derive the transfer functi on, the dc gain and the 3-dBfrequency. Design the circuit to ob tain an input resistance of 1k Ω, a dc gain of 20dB, and a 3dBfrequency of 4kHz. At what frequency does the magnitude of the transf er function reduce tounity?

4. (10%) Analyze the circuit shown in Fig. 4 to determine the voltages at all nodes and the currentsthrough all branches. Let V t = 1V and k 'n(W/L) = 1 mA/V 2. Neglect the channel-lengthmodulation effect (i.e., assume λ=0). (10%)

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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