課程名稱︰線性代數
課程性質︰Orz-stO
課程教師︰逢世邁
開課系所︰電機系
考試時間︰2006.3.15
試題 :
1.Let the matrix A and the vector b be respectively defined by
[ 1 1 -1 -2 -8] [-3]
[-2 -2 1 2 9] [ 5]
A=[ 3 3 -2 -3 -15] b=[-9]
[ 1 1 -1 -1 -6] [-4]
(a) Let [R c] be the reduced row echelon form of [A b].Find [R c].
(b) What are the rank and nullity of A?
(c) Is the equation Ax=b consistent? If it is, find the parametric
representation of the general solution.
(d) Find the parametric representation to the equation Ax=0.
(e) Find a linearly independent subset of R^5 that spans the null space
of Ta.
(f) Select as many vectors as possible from the columns of A to form
a linearly independent set.
2. (a) Determine if the following matrix is invertible. If it is,find its
inverse.
[ 0 1 1 1]
[a1 a2 a3 a4] = [ 1 0 1 1]
[ 1 1 0 1]
[ 1 1 1 0]
(b) Let T: R^4 => R^4 be a linear transformation such that
T(a1)=a2 T(a2)=a3 T(a3)=a4 T(a4)=a1
Is T uniquely determined by the four images above? If it is, find
its inverse.
[x1] [ x1+x2+x3 ]
3. Let T: R^3 => R^3 be defined by T([x2]) = [2*x1+x2+x3]
[x3] [ x1-x3 ]
(a)Find the standard matrix A.
(b)Is T onto? Is T one-to-one?
(c)Find a spanning set for the null space of A.
(d)Find a linearly independent set for the range of T.
4.Let A be a 4*3 matrix and B a 3*4 matrix.
(a)Is it possible that Ax=b is consistent for every b in R^4? Explain
your answer.
(b)Is it possible that Bx=c has a unuque solution for some c in R^4?
Explain your answer.
5. Let S1={u1 , u2 , u3... ,uk} be a subset of R^n and S2={v1,v2...vk}
be a subset of SpanS1.
(a)Let A=[u1 u2 u3...uk] and B = [v1 v2 ...vk]. Prove that if S2 is
linearly independent, then there exists an invertible matrix C
such that B = AC.
(b)Prove that if S2 is linearly independent, then S2 is also linearly
independent.
(c)Is it true that if S1 is linearly independent, then S2 is also
linearly independent? Justify your answer.
6. (a) Let A be an n*n matrix such that A^2-3A+2I = O. Show that A is
invertible and find a general expression for A inverse (express
your answer in terms of A).
(b) Let B be a 2*2 matrix such that B^2=I. Does this mean that
[±1 0 ] [±1 0 ]
B=[0 ±1]? If yes,prove it. If no,find such B≠[0 ±1] but
B^2 = I.
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