112年 - 112 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#147401

(a) \( (12\%) \) Compute the LU factorization of A.

(b) \( (12\%) \) Find the bases for the four fundamental subspaces of \( \mathbf{A} \), namely, the nullspace \( N(\mathbf{A}) \), the nullspace \( N(\mathbf{A}^{\mathrm{T}}) \), the range \( R(\mathbf{A}) \), and the range \( R(\mathbf{A}^{\mathrm{T}}) \), respectively.

(c) \( (6\%) \) Verify that \( \mathbf{A} \) satisfies the fundamental subspaces theorem.

(a) \( (10\%) \) Show that \( L \) is a linear transformation.

(b) (10%) Give the following two ordered bases: \( E = \{\mathbf{u}1, \mathbf{u}2\} \) where \( \mathbf{u}1 = (2, 1)^\mathrm{T} \), \( \mathbf{u}2 = (1, 3)^\mathrm{T} \), and \( F = \{\mathbf{v}1, \mathbf{v}2, \mathbf{v}3\} \) where \( \mathbf{v}1 = (1, 0, 0)^\mathrm{T} \), \( \mathbf{v}2 = (1, 1, 0)^\mathrm{T} \), \( \mathbf{v}3 = (1, 1, 1)^\mathrm{T} \). Find the matrix representation of \( L \) with respect to the ordered bases \( E \) and \( F \).

(a) \( (10\%) \) Solve the least squares problem \( \mathbf{B}\mathbf{x} = \mathbf{b} \) for \( \mathbf{b} = (1,3,2,4)^{\mathrm{T}} \)

(b) \( (10\%) \) Compute the Gram-Schmidt QR factorization of the matrix B.

4. \( (20\%) \) Diagonalize the matrix \( \mathbf{C} = \begin{bmatrix} 2 & 0 & 0 \\ -1 & 4 & 0 \\ -3 & 6 & 2 \end{bmatrix} \).

5. (10%) Let \( \mathbf{D} \) be an \( m \times n \) matrix of rank \( n \). Is the matrix \( \mathbf{D}^{\mathrm{T}}\mathbf{D} \) symmetric positive definite? Explain your answer.