92年 - 92 淡江大學轉學考線性代數#56121

科目：轉學考-線性代數 | 年份：92年 | 選擇題數：0 | 申論題數：11

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所屬科目：轉學考-線性代數

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申論題 (11)

【已刪除】1. Let A=

• Find det(A). (10 points)

2. LetT: R₃ → R₂ be a linear transformation and

T(l，0，0)=(l，-2)，T(l,l,0)=(l,3), T(0,0，l)=(2，-1). Find

T(x，y，z). (10 points)

(a) Find general solutions of AX=0. (10 points)

(b) Find Rank(A). (5points)

【已刪除】4. Let P=

. Show that P is invertible and find p^-1 .(10 points)

5. Let T : P₂ → R² be deflned by T(a+bx+cx²) = (a-b,c+a)

and B={1, x,x²}，D={(1,-1)，(0，1)}. Find the matrix of T corresponding to the ordered bases B and D. (10 points)

6. Suppose that {x, y} is a linearly independent set in a vector space V. Show that if T : V -＞ W is a" one -to-one linear transformation, then {T(x+2y), T(2x-y)} is also linearly independent. (10 points)

(a) Find the characteristic polynomial of A. (5points)

(b) Find an invertible matrix P such that p^-1 AP is diagonal. (10 points)

8. Given line L: x=z-y=y-z in R³, let P(x,y,z) be the orthogonal projection of (x,y,z) on L. Find P(x,y,z). (10 points)

9. Let T : V → W be a linear transformation, where V and W are finite dimensional vector spaces. Show that T is onto if and only if there exists a linear transformation S : W-» V such that TS(w)= w for all w in W. (10 points)