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(MATH113)[2011](f)midterm~rfarera^_58362.pdf
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Math2111 Introduction to Linear Algebra
Fall 2011
Midterm Examination (All Sections)

Name:
Student ID:
Lecture Section:

.
There are FOUR questions in this midterm examination.

.
Answer all the questions.

.
You may write on both sides of the paper if necessary.

.
You may use a HKEA approved calculator. Calculators with symbolic calculus functions are not allowed.

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The full mark is 100.


1 42
.. ....
.. ....

1. Let a=2, a=5, a=1 be the three columns of matrix A=[aaa].
23 3
1 .. .... 12 ..360
....
.. ....
(a) Determine if the homogeneous system A=has a nontrivial solution. If there is a
x0
nontrivial solution, then write down the solution set in parametric vector form. [8]
(b)
Is the set of columns of A, {1, 2, 3}

aaa, linearly independent or linearly dependent? [2]

(c)
Is the linear system A= consistent for each vector bin R ? If the answer is yes,


xb3
.. ..
b1
bbbin for
explain why. Or if the answer is no, find out the condition on 1, 2, 3 b=b2
.. ..
b3
.. A= being consistent. [5]
xb
(d) Do the columns of Aspan R3 ? If not, what is the geometric description of Span{1, 2, 3}
aaa? [3]
3
(e)
Consider the linear transformation x6 Ax. Is it onto R ? Is it one-to-one? [4]

(f)
Is the matrix A invertible? If the answer is yes, find out A.1 . [3]


[25] in total
Solution:
(a) Row operations on the augmented matrix yield the reduced echelon form
.1420.. 14 20.. 10 .20..x1 =2t ... .. . .
2510 0 .3 .30 0 1 1 0 , from which we have .x=.tand
... .. . 2
.
.3600.. 00 00.. 00 0 0. x3 =t
... .. . .
2
..
..

x=.1 t. Therefore, there are nontrivial solutions.
..
..
1
..
(b) Since A= has nontrivial solutions, {aaa, , is linearly dependent.
x0}
123
(c) Row operations on the augmented matrix yield .142 b.. 14 2 b .. 14 2 b .
11 1
... .. .
251 b2 0 .3 .3 b2 .2b1 0 .3 .3 b2 .2b1 , from which we
... .. . . 60 3 .. 0 .6 .63 .b1 .. 000 3 .2bb 1 .
3 bb 3 b 2 +.
. .. ..
see that the system is not consistent for each b. The condition for the system to be consistent is b.2bb
+=0.
3 21
3

(d) No, the columns of Adont span R. Span {aaa , , }is geometrically a plane
123
according to the consistency condition b2 += found in (c).
.bb0
3 21
(e) It is not onto R3 as the system Axb
=is not always consistent. It is not one-to-one either as there are nontrivial solutions for A=.
x0
(f) The matrix Ais not invertible.
2. Let A = aaaaaa] be a 464
[ matrix and the columns of A span R.
123456
(a)
How many pivot positions are there in A? [4]

(b)
Is the set { , , , , , }


aaaaaa linearly independent or linearly dependent? [4]
123456
(c) Does the homogeneous system A =have a nontrivial solution? If the answer is
x0
yes, then find out the number of free variables in the solution set. [5]
(d) For each vector bin R , is the linear system A = always consistent? [4]
4