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

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 \( A^T A \vec{v} \cdot \vec{v} = A \vec{v} \cdot A \vec{v} = 0 \))