(c) Which matrices \( \mathbf{M} \) have \( \mathbf{S} \) as their column space?

(d) Give an invertible \( n \) by \( n \) matrix \( \mathbf{A} \). Show that \( \mathbf{A}\mathbf{v}1, \mathbf{A}\mathbf{v}2, \) and \( \mathbf{A}\mathbf{v}_3 \) is also a basis for \( \mathbf{S} \).

6. (10%) Factor the matrix B into B = LDLᵀ.$$\mathbf {B} = \left[ \begin{array}{c c c} 5 & 5 & 0 \\ 5 & 7 & 2 \\ 0 & 2 & 9 \end{array} \right]$$

7. (10%) Project \( \mathbf{b} = \begin{bmatrix} 6 \\ 7 \\ 8 \\ 9 \end{bmatrix} \) onto the column space of \( \mathbf{C} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \). What is the projection matrix \( \mathbf{P} \)? What is the projection \( \mathbf{p} \)?

(a) Are all the pivots of \( \mathbf{D} \) positive?

(b) Are all the upper left determinants of \( \mathbf{D} \) positive?

(c) Are all the eigenvalues of \( \mathbf{D} \) positive?

(d) Is \( x^{T}Dx \) always positive except at \( x = 0 \)?

(e) Does \( \mathbf{D} \) equal \( \mathbf{R}^{\mathrm{T}}\mathbf{R} \) for a matrix \( \mathbf{R} \) in which all the columns are independent?

III、Short Essay (20%)Please read the paragraph below and respond to the associated questions. Consider offering detailed and thoughtful answers.As a cofounder and the CEO of the U.S. chipmaker Nvidia, Jensen Huang operates in a high-velocity industry requiring agile, innovative thinking. Reflecting on how his leadership style has evolved, he told the New York Times, “I probably give fewer answers and I ask a lot more questions...It’s almost possible now for me to go through a day and do nothing but ask questions.” He continued, “Through probing, I help [my management team]...explore ideas that they didn’t realize needed to be explore(D)” (quoted from The Art of Asking Smarter Questions, Harvard Business Review, May–June 2024.)Based on the above description and what is known from the media about Jensen Huang, (1) please analyse Huang’s leadership style to align with key leadership theories and (2) explain its potential impact on innovation and team development.

IV、Short Essay (20%) In August 2024, 13 current and former TSMC(台積電)employees filed a class action lawsuit in Californi(A) The 13 plaintiffs, whose backgrounds include the US, Mexico, Nigeria, Europe and South Korea, were seeking damages to redress TSMC’s apparent discrimination practices. In the lawsuit, the plaintiffs claimed TSMC’s employment practices include an “intentional pattern and practice of employment discrimination against individuals who are not of East Asian race, not of Taiwanese or Chinese national origin, and who are not citizens of Taiwan or China, including discrimination in hiring, staffing, promotion, and retention/termination decisions.” While declining to comment on the lawsuit, TSMC did say that recruitment and promotion in the company did not take into account factors such as nationality and rac(E) “TSMC believes strongly in the value of a diverse workforce, and we hire and promote without regard to gender, religion, race, nationality, or political affiliation because we respect differences, and believe that equal employment opportunities strengthen our competitiveness,” the company sai(D) “We also provide various channels for employees to raise concerns, and strive to address concerns constructively,” it sai(D)(Source: Taipei Times, Friday Nov 15, 2024) Imagine you are a consultant hired by TSMC to improve its diversity and inclusion efforts. Based on the lawsuit allegations, list, explain and identify potential issues with TWO specific recommendations that TSMC could implement in its organizational structure and processes to mitigate unconscious bias in hiring, promotion, and performance evaluation.

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