106年 - 106 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_資訊工程學系3年級：線性代數#147406

1. A matrix \( \mathbf{M} \) is called a skew symmetric if \( \mathbf{M}^{\mathrm{T}} = -\mathbf{M} \). Prove that if \( \mathbf{A} \) is an \( n \times n \) skew symmetric matrix and \( n \) is odd, then \( \mathbf{A} \) is singular. (12%)

2. Find the standard matrix representation for the following linear transformation L: L reflects each input vector x about the line \( x_2=x_1 \) and then projects it onto the \( x_1 \) axis.(12%)

3. Let M be a \( 2 \times 2 \) matrix with \( \text{tr}(M)=\sqrt{3} \) and \( \text{det}(M)=-2 \). Let \( \alpha_1 \) and \( \alpha_2 \) be eigenvalues of M. What is \( |\alpha_1-\alpha_2| \) ? (12%)

4. Find a matrix \( \mathbf{B} \) such that \( B^3 = A \) where \( \mathbf{A} = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 3 \\ 1 & 1 & -1 \end{bmatrix} \) (14%)

(1)Find the 3 by 3 projection matrix \( P \) onto the plane of \( a_1 \) and \( a_2 \). (5%)

(2)Find the projection \( p \) of \( b \) onto the plane of \( a_1 \) and \( a_2 \). (5%)

(3) Find the error vector \( e = b - p \). (5%)

(4)Find a vector whose projection onto the plane of \( a_1 \) and \( a_2 \) is the zero vector. (5%)

(1) Find the column space of A. (5%)

(2) Find the complete solution of \( \mathrm{Ax} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \). (10%)

(3) Give a basis for the nullspace of A. (5%)

7. Invert the matrix $$\mathbf{B} = \begin{bmatrix} 2 & 2 \\ 0 & 2 \end{bmatrix}$$ by the Gauss-Jordan method starting with [B I]. (10%)