108年 - 108 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#147395

1. (10%) Find the reduced row echelon form of the matrix $$\begin{bmatrix} -1 & 1 & -1 & 3 \\ 3 & 1 & -1 & -1 \\ 2 & -1 & -2 & -1 \end{bmatrix}$$

2. (10%) Use the elementary row operations to find the inverse of the matrix $$\begin{bmatrix} 2 & 0 & 5 \\ 0 & 3 & 0 \\ 1 & 0 & 3 \end{bmatrix}$$

3. (10%) Let \( A=[\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3] \) be a 3x3 matrix and suppose that \( 2\mathbf{a}_1+\mathbf{a}_2-4\mathbf{a}_3=\mathbf{0} \). How many solutions will the system \( A\mathbf{x}=\mathbf{0} \) have?

4. (10%) Let $$x1 = \begin{bmatrix} 1 \\ -1 \\ 2 \\ 3 \end{bmatrix}$$, $$x2 = \begin{bmatrix} -2 \\ 3 \\ 1 \\ -2 \end{bmatrix}$$, and $$x_3 = \begin{bmatrix} 1 \\ 0 \\ 7 \\ 7 \end{bmatrix}$$. Determine whether the vectors are linearly independent.

5. (10%) Show that $$e^x,e^{-x}$$, $$e^{2x}$$ are linearly independent in C[0,1] where C[0,1] is the set of continuous functions defined on [0,1].

(a)(6%) Find the bases for $$N(A^T)$$, namely, the null space of $$A^T$$.

(b)(6%) Find the bases for $$R(A)$$, namely, the column space of A.

(c)(6%) Show that $$N(A^T)$$ and R(A) are orthogonal complement.

(a)(6%) Determine whether the mapping L is a linear transformation.

(b)(6%) Find a matrix A to perform the mapping $$L(x)$$.

(a)(6%) Find the eigenvalues and the corresponding eigenvectors for A.

(b)(6%) Diagonalize A as $$A = XDX^{-1}$$ where D is a diagonal matrix.

9. (8%) Prove that if \( \mathbf{A} \) is an \( n \times n \) orthogonal matrix then \( \mathbf{a}_1 \mathbf{a}_1^{\text{T}} + \mathbf{a}_2 \mathbf{a}_2^{\text{T}} + \cdots + \mathbf{a}_n \mathbf{a}_n^{\text{T}} = \mathbf{I} \), where \( \mathbf{a}_i \) is the \( i \)th column of \( \mathbf{A} \), and \( \mathbf{I} \) is the identity matrix.