107年 - 107 東吳大學_碩士班招生考試_數學系(A組)：線性代數#101986

科目：研究所、轉學考(插大)◆線性代數 | 年份：107年 | 選擇題數：0 | 申論題數：8

試卷資訊

所屬科目：研究所、轉學考(插大)◆線性代數

選擇題 (0)

申論題 (8)

1. Use the following linear system to verify that the general solution of a consistent linear system
Ax=b can be obtained by adding any specific solution of Ax=b to the general solution of Ax=0.
x + 2 y − 3z = 3
3 x + 6 y − 9z = 9
−2 x − 4 y +6 z = −6

2. (a) Find the standard matrix for the transformation T(x₁, x₂, x₃)=(x₁-2x₂, x₂+3x₃, 2x₁- x₃)in R³.

(b) Find the standard matrix for the reflection about the line y=x in R².

(c) Suppose T is a linear transformation from R² to R², and transforms (1,2) to (-3,4) and (2,0) to (1,5). Find T(v), T²(v) for v=(3, -2).

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

(b) Determine whether the set W of all 2 × 2 symmetric matrices is a vector subspace of . Is it a finite-dimensional vector subspace?

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

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 ] .