105年 - 105 國立臺北大學_日間學士班曁進修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#147403

1. (10%) Let $$u1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, u2 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, u3 = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, b1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, b2 = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$$. Let L be the linear transformation defined by $$L(x1, x2, x3) =$$

$$(x1+x2, x1-x3)$$. Find the matrix A representing L with respect to the bases $$\{u1, u2, u3\}$$ and $$\{b1, b_2\}$$.

2. (12%) Find the least squares solution of the following system:

$$\begin{array}{l} x1 + x2 + x3 = 4 \\ -x1 + x2 + x3 = 0 \\ -x2 + x3 = 1 \\ x1 + x3 = 2 \end{array}$$

3. (12%) Find the eigenvalues and the corresponding eigenspaces for the following matrix:

$$A = \begin{bmatrix} 3 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$$

4. (16%) Factor the matrix A into a product $$XDX^{-1}$$, where D is diagonal and $$A = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 3 \\ 1 & 1 & -1 \end{bmatrix}$$.

(4%) (a) X, (4%) (b) D, (4%) (c) $$X^{-1}$$. (4%) (d) Find a matrix B such that $$B^3=A$$

5. (18%) Compute L and U for the following matrix A:

$$A = \begin{bmatrix} 4 & 4 & 4 & 4 \\ 4 & 5 & 5 & 5 \\ 4 & 5 & 6 & 6 \\ 4 & 5 & 6 & 7 \end{bmatrix}$$

6. (16%) Find the nullspace of the following matrix B:

$$B = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

7. (16%) Compute the determinant of the following matrix C:

$$C = \begin{bmatrix} 1 & 12 & 0 & 0 \\ 0 & 14 & 0 & 0 \\ 0 & 16 & 1 & 0 \\ 0 & 18 & 0 & 1 \end{bmatrix}$$