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

Name:
Student ID:
Lecture Section:

.
There are SEVEN questions in this final examination.

.

.
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.

.
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).



(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).

(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).

(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).



(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).


 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 .

(b) For the matrix A given in (a), what is the dimension of the null space of A .

(c) For the matrix A given in (a), what is the dimension of the null space of AT .

 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

(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 .

 in total
Solution:

..16.
4. Let A=.
..
.14
..
(a) Find all the eigenvalues of A.

234
(b) Evaluate BIAA A Awhere I identi