108年 - 108 國立臺北大學_碩士班一般入學考試試題_資訊工程學系：線性代數與離散數學#147261

(a) How many words begin with R and end with T?

(b) How many words begin with A or end with B?

(c) How many words begin with A or R or end with T or B?

2. (20%) Consider the following graph. It has 20 vertices each with degree 3. Answer the questions. Is it a planar graph? Does it have an Euler circuit? Does it have an Euler path? Does it have a Hamilton circuit? Does it have a Hamilton path? Prove all your answers.

3. (15%) Boolean function \( f(x, y, z, w) = -x \vee (x \wedge y \wedge z \wedge w) \vee (x \wedge y \wedge z \wedge -w) \). First, find the truth table of function \( f \). Second, simplify the expression of \( f \) to minimize the number of AND, OR, NOT operations. Third, draw out the combinatorial circuit corresponding to your simplified expression. Finally, transform and redraw your combinatorial circuit to contain NAND gates only.

4. (15%) Let \( \mathbf{R} \) denote the set of real numbers. Show that \( \mathbf{R} \), together with the usual addition and scalar multiplication of real numbers, is a vector space.

(a) Show that the mapping \( L \) is a linear transformation.

(b) Determine the kernel of \( L \)

(a) Find the eigenvalues and eigenvectors of the matrix \( \mathbf{A} \).

(b) Diagonalize the matrix \( \mathbf{A} \).

(c) Compute \( \mathbf{A}^5 \)