89年 - 89 淡江大學轉學考線性代數#56164

科目：轉學考-線性代數 | 年份：89年 | 選擇題數：0 | 申論題數：13

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所屬科目：轉學考-線性代數

選擇題 (0)

申論題 (13)

【已刪除】(a) Prove that is a vector space over

. (5%)

(b) Give a basis for A4 and find the dimension of M, (5%)

【已刪除】2. Let A be an n x n matrix with real entries. Let W

be the null space of A, and let c

be a particular solution to the system AX = B. Prove that c + W is the complete set of solutions to AX = B. (10%)

【已刪除】3. Let T : U→ V and S : V →W be linear transformations. Prove that the

【已刪除】 (a) If W and U are both subspaces of a vector space V, then

is also a subspace of V. (5%)

【已刪除】(b) If W and U are both subspaces of a vector space V y then

is also a subspace of V (5%)

(c) det(A + D) =det(A)+dct(B), where clet(A) denotes the determinant of a square matrix A. (5%)

(a) Find the matrix representations of S、T and T o S relative to the standard basis. (6%)

(b) Find the matrix representations of S relative to the basis {(1，1)，(0, 2)}. (9%)

(c) Evaluate S¹⁰⁰(1,—1). (10%)

(a) Find the characteristic polynomial and the minimal polynomial of A. ( 10%)

(b) Find the eigenvalues and the corresponding cigcnspaces of A. (10%)

(c) Find the Jodon canonical form J for A, and find a matrix P such that P^-1 AP = J. (10%)