103年 - 103 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：線性代數#145976

(a) Find the characteristic polynomial and the minimal polynomial of .(6分)

(b) Find all eigenvalues and eigenvectors of .(6分)

(c) Find an invertible matrix such that is diagonal and use it to find . (8分)

2. Let be an real matrix, be an real matrix. Prove that (15分)

(a) Evaluate the determinant .(8分)

(b) Find the inverse matrix .(7分)

4. Suppose \( T(v_1) = w_1 + w_2 + w_3 \), \( T(v_2) = w_2 + w_3 \), and \( T(v_3) = w_3 \). What coefficients \( \alpha, \beta, \gamma \) gives \( T(\alpha v_1 + \beta v_2 + \gamma v_3) = w \). (10 分)

5. Let \( u, v \) and \( w \) be distinct vectors of a vector space \( V \). Show that if \( \{u, v, w\} \) is linearly independent, then \( \{u+v, u+v+w, v+w\} \) is also linearly independent. (10 分)

6. 6. Fill in the five entries: ^a,b,c,d,e in the following 4x4 matrix

so that the matrix is symmetric.(15分)

7. For vectors \( ^{v} \) and \( ^{w} \) in an inner product space \( \langle \cdot, \cdot \rangle \). Prove that \( \langle v-w, v+w \rangle = 0 \) if and only if \( \langle v,v \rangle = \langle w,w \rangle \) (15分)