課程名稱︰線性代數
課程性質︰必修
課程教師︰程舜仁
開課系所︰數學系
考試時間︰
試題 : Final Exam
Date:January 11,2006
Course Name:Linear Algebra
Course Instructor:Shun-Jen Cheng
Instructions: You need to justify your answers in order to receive full
credits. You may use ANY result proved in class. Good luck!
1.(10 points)Find the inverse of the matrix:
┌ ┐
│1 1 1 1│
A=│1 -1 0 -2│
│1 1 0 4│
│1 -1 0 8│
└ ┘
2.(10 points)Solve the following system of linear equations:
(x y z w u stands for variables)
2x + 3z - 4u = 5
3x - 4y + 8z - 3w = 8
x - y + 2z + w - u = 2
-2x + 5y - 9z - 3w -5u = -8
3.(10 points)Find all solutions to the homogeneous linear differential
equation
y^(6) - 2y^(3) + y = 0
in the space C^∞(R, C)
4.(10 points)Let A be an m × nmatrix with rank m. Prove that there exists
n × m matrix B such that AB = I (m × m identity matrix)
5.(15 points)Let A be an n × n matrix such that A^m = 0, for some m. Prove
that the matrix I(n × n) + A is invertible.
6.(15 points)Let X1, X2, ...., Xn be any numbers. Let
┌ ┐
│1 1 ... ... 1 │
│X1 X2 ... ... Xn │
│X1^2 X2^2 ... ... Xn^2 │
A = │ . . . . . │
│ . . . . . │
│ . . . . . │
│X1^(n-1) X2^(n-1)... ... Xn^(n-1)│
└ ┘
Prove that detA = Π . Hint: use induction on n.
1≦i<j≦n(Xj-Xi)
7.(15 points)Let V be a finite-dimensional vector space. Let T : V → V be a
linear map. Prove that for every i 屬於自然數 we have
2dim(Kernel T^(i+1)) ≧ dim(Kernel T^(i+2)) + dim(Kernel T^(i)).
8.(15 points)Let V and W be finite-dimensional vector spaces and T : V → W be
a linear map. Prove that T : V → W is one-to-one if and only if
T* : W* → V* is onto.
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