Linear Algebra

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2012–13 First Semester MATH 1111 Linear Algebra Chapter 1: Matrices and Systems of Equations

Coverage of Chapter 1:   Skip Application 3 in Section 1.4. Skip ‘Triangular Factorisation’ in Section 1.5.

A.

Solving Equations

1.

We are all familiar with solving equations. Illustrate how the following equations can be solved, and then raise and answer some theoretical and/or practical questions concerning the process of solution. (a) 3x  1  8

2 x  x  1 (b)  1 2  3 x1  x2  4

(c) (d)

x 4  5 x 2  36  0
x2  x4

B.

Systems of Linear Equations

(Ref: Sections 1.1 and 1.2)

2.

Your first experience of solving systems of equations was probably to deal with a system of two equations in two unknowns. Can such equations always be solved? Are there any special cases?

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3.

The system mentioned in the previous question is usually called a 2  2 system. How about n  n systems in general? (Note: An m  n system is a system consisting of m __________ and n __________.)

4.

Solve the following systems of equations. (a)
 x1  2 x2  x3  3    x2  x3  2  2 x3  8   x1  2 x2  x3  3  (b)  3 x1  x2  3 x3  1  2 x1  3 x2  x3  4

5.

Look at the previous question again and answer the following questions. (a) Which system is easier to solve? Why? (b) We say that the two systems are equivalent. Explain. (c) Identify three operations in the process of solving for the systems. (d) A system of linear equations can be represented by an augmented matrix. Illustrate with the systems in the previous question. (e) How do the operations in (c) correspond to row operations on the corresponding augmented matrices?

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6.

A matrix is said to be in row echelon form if it satisfies the following conditions:    The first non-zero entry of a non-zero row must be ___, called a ______________. Except for the first row, any leading 1 must be on the ________ of the leading 1 in the previous row. Zero rows, if any, must be ___________________________________.

It is said to be in reduced row echelon form if it furthermore satisfies the following:  Any leading 1 must be the only ___________________ in the column.

7.

Which of the following matrices are in row echelon form? Which of them are in reduced row echelon form? (a) 1 0 0 0   0 1 0 0 0 0 1 1   1 0 1 0 (b)  0 1 1 0    0 0 0 1  

(c) 1  2 3  0

0 1 1 0   0 0 1 0 1 0 1 1  

1  0 (d)  0  0

1 2 0 0

0 0 0 0

1 2 3 0

1  2 3  4

(e)

0  0 0  0

1 0 0 0

1 2 0 0

1 2 0 0

(f)

1  0 0  0

0 1 1 0

0  1 0  1

8.

How can we transform a matrix into 1 1 1 Illustrate with the matrix  2 1 3   2 1 5 

reduced row echelon form by means of row operations? 1  1. 2 

(Note: The process of using row operations to transform a matrix into row echelon form is usually called _________________________, while that of transforming a matrix into reduced row echelon form is usually called ________________________.)

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9.

(a) How are row operations related to solving systems of linear equations? (b) Explain the meaning of (i) lead variables

(ii) free variables (iii) (in)consistent systems

10. An m  n system is said to be (a) an overdetermined system if ___________ (b) an underdetermined system if ___________ (c) a homogeneous system if _______________________________________________

11. Show that (a) a homogeneous system is always consistent; (b) an underdetermined homogeneous system always has a non-trivial solution.

C.

Operations on Matrices

(Ref: Sections 1.3 and 1.4)

12. Give some examples in daily life in which matrices can be useful.

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13. Explain the meaning of the following terms and notations. (a) scalar (b) vector / row vector / column vector (c) n

(d) m  n matrix (e) (i, j)-entry (f) square matrix

(g) diagonal matrix (h) upper/lower triangular matrix

14. (a)...
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