(b) Give the space \( \mathbf{S} \) which they span. What is the dimension of \( \mathbf{S} \)?

(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.

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