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(MATH2111)[2011](f)final~=hneh5he^_82709.pdf
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Math2111 Introduction to Linear Algebra
Fall 2011
Final Examination (All Sections)

Name:
Student ID:
Lecture Section:

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There are SEVEN questions in this final examination.

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

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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. Short questions:
(a)
If the set of the columns of a square matrix is linearly independent, then the set of the rows of the same matrix is also linearly independent. Answer True or False (no reason required).

[4]

(b)
If the three vectors vvv, , in a vector space Vform a linearly dependent set and

123
Tis a linear transformation from the vector space Vto a vector space W, then (v), ( Tv Tv) also form a linearly dependent set. Answer True or False (no
T ), (
123
reason required).
[4]
(c) Let Aand Bdenote two nn square matrices. If AB is an invertible matrix, then A

and B are both invertible matrices. Answer True or False (no reason required).
[4]
(d)
Let T: Rn Rmbe a linear transformation. What is the dimension of the range of T if T is a one-to-one transformation. Write down the answer (no explanation required).

[4]

(e)
Let T:R n Rmbe a linear transformation. What is the dimension of the kernel of Tif T maps Rn onto Rm. Write down the answer (no explanation required).

[4]
[20] in total
Solution:
(a)
True.

(b)
True.

(c)
True.

(d)
nA= n. dim Nul =. 0 = n. Here A is the

. By the Rank theorem, dim Col An standard mn matrix of T.
.. dim Col A nm
(e) nm. By the Rank theorem, dim Nul A= n =. . Here A is the standard mn matrix of T.
2.
.2 .24 .2.
..

.33 .63
..
(a) For the matrix A =.6 .39 3 ., use a subset of the five rows of A to form a ..
2 .35 .4
..
.5 .49 1 .

..
basis for the row space of A .
[8]
(b) For the matrix A given in (a), what is the dimension of the null space of A .
[1]
(c) For the matrix A given in (a), what is the dimension of the null space of AT .
[1]
[10] in total
Solution:

(b) rank A =rank AT =3 . By the Rank Theorem, dim Nul A =43
.=1.
(c) rank A =rank AT =3 . By the Rank Theorem, dim Nul AT =53.=2.
3. .p ()t =+5tt2
.12.p2()t =. . 18t 2t
(a) Given the four vectors . in the vector space P of polynomials of .p3()t =.+ + 34t 2t22
.p ()t =.23t
.4 degree at most 2, determine if the set {p (), p (), t p (), p()t }spans P.
tt
1234 2
[7]
(b) Is the set {p1(), t p2 t p3(), 4 }
(), t p() t linearly independent or linearly dependent? If it is linearly dependent, then find one vector pj ()t (with j >1 ) which is a linear combination of the preceding vectors, p1()t , , p j.1()t .
[3]
[10] in total
Solution:

..16.
4. Let A=.
..
.14
..
(a) Find all the eigenvalues of A.
[5]
234
(b) Evaluate BIAA A Awhere I identi