110年 - 110東吳大學_碩士班招生考試_數學系︰線性代數#100057

科目：研究所、轉學考(插大)◆線性代數 | 年份：110年 | 選擇題數：0 | 申論題數：8

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所屬科目：研究所、轉學考(插大)◆線性代數

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申論題 (8)

1. (a) In R³, represent all the linear combinations of v=(2,0,1) and w=(0,1,2), give a geometric interpretation. And find a vector which is not a combination of v and w.

(b) In the following two planes, which one is a subspace of R³ (give your reasons) and find a basis for the subspace. x +2y-3z -4=0, 2x -4y +5z =0

2. (a) If A is a 3 × 3 matrix has the determinant det(A) = -5, then what is det and det ?

(b) Let E be an n × n elementary matrix, what is the determinant det(E) of E ?

(c) 8% For the matrix , find the value of , where is the cofactor of entry

3. For , find the reduced row echelon form of A and verify the dimension theorem for matrices.

4. Find the eigenvalues and eigenvectors of the matrix . Is A diagonalizable? And find the Jordan form of A.

5. Let R³ have the Euclidean inner product, and let v₁ = (1, 0, 0), v₂ = (1, 0, -1), v₃ = (1,1, 0) . Use the Gram-Schmidt process to find the QR-decomposition of A =[ v₁ v₂ v₃ ] .