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(math111)[2008](f)MATH111Q1~PPSpider^_10427.pdf
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MATH111 Quiz 1 Solutions

1. (4 marks) Describe all the solutions of the following system in parametric form:
. .

x1 . x2 +2x3 . x4 =1
.
2x1 . x2 . x3 . x4 =1
.
.
x1 . 3x3 . x4 = .1
Solution: The solution set of the system is
.
.

x1 =3s
.
x2 = .2+5s
.

x3 = s
.
x4 =1
where s is a free parameter.
2. Determine whether the following statements are true. If true, justify your answer; if false, give a counterexample.
(a) (2 marks) If the linear system Ax = b has more than one solution, then so does the homogeneous system Ax = 0. True. First of all, Ax = 0 is always consistent, because it has the trivial solution. Denote x1,x2 as two di.erennt solutions of Ax = b. It follows
that x1 . x2 is a nonzero solution of Ax = 0. Therefore Ax = 0 has at least
two di.erent solutions namely x1 . x2 and the trivial solution.
(b) (2 marks) If a system of linear equations has two di.erent solutions, then it must have in.nitely many solutions. True. Denote the linear system as Ax = b, and let its two di.erent
solutions be x1,x2. Then for an arbitrary real number k, kx1 + (1 . k)x2 is a solution of Ax = b.
(c) (2 marks) Suppose u, v, w are vectors such that {u, v}, {v, w}, {w, u} are
all linearly independent sets. Then so is {u, v, w}.
.. .. .. .. ..
110
. .. .. .
False. For example, let {u, v, w} = ,, .
..
011
1

3. (4 marks) Determine h, k and s such that: (i) is inconsistent; (ii) is linearly independent.
. .. .. .. . ..
x1 +3x2 = k 1 s
... .
(i)(ii) ,
. ..
4x1 + hx2 =8 ss +2
... .
13 k 13 k
Solution: (i) .. ... It is now obvious to see 4 h 80 h . 12 8 . 4k
(a)
If h = 12,k .

= 2, it is inconsistent;

(b)
If h = 12,k = 2, it has in.nitely many solutions;

(c)
If h .


= 12, it has a unique solution.
(ii) It su.ces to choose proper s such that Ax = 0 has a unique solution, where
... .
1 s 1 s
... .
A = . Now A . It is obvious that it has a ss +2 0(2 . s)(1 + s)
unique solutions if and only if s ..
=2,s = .1.
4. (4 marks) Let T : R3 R3 be a linear transformation. Assume T maps R3 onto R3 . Show that T must be one-to-one. (Hint: use the equivalent statements in terms of the standard matrix A.) Proof. Let 3 3 matrix A be the standard matrix of T . Since T maps R3 onto R3, each row of A has a pivot position. In other words, A has 3 pivot positions. Hence, each column of A has a pivot position, which implies the columns of A are linearly independent. Therefore, T is one-to-one.
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