2. Discuss the solution of a linear system which has unknown variable
§1.1 Systems of Linear Equations
Existence and Uniqueness Questions
Two fundamental questions about a linear system 1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is it the only one; that is, is the solution unique?
augmented matrix
a11 a12

a21
a22



am1 am2
a1n
a2n


amn

a11 a12

a21
a22


am1 am2
a1n b1
a2n
b2


amn
bm

§1.1 Systems of Linear Equations
Theorem 1 Uniqueness of the Reduced Echelon Form
Each matrix is row equivalent to one and only one reduced echelon matrix.
§1.2 Row Reduction and Echelon Forms
The Row Reduction Algorithm
Step1 Begin with the leftmost nonzero column. Step2 Select a nonzero entry in the pivot column as a pivot. Step3 Use row replacement operations to create zeros in all positions below the pivot. Step4 Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify. Step5 Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot.
Linear Algebra and Its Application
REVIEW FOR THE FINAL EXAM
Gao ChengYing
Sun Yat-Sen University Spring 2007
REVIEW FOR THE FINAL EXAM
Chapter 1 Linear Equations in Linear Algebra Chapter 2 Matrix Algebra Chapter 3 Determinants Chapter 4 Vector Spaces Chapter 5 Eigenvalues and Eigenvectors Chapter 6 Orthogonality and Least Squares Chapter 7 Symmetric Matrices and Quadratic Forms
consistent
§1.1 Systems of Linear Equations
Solving a Linear System
Elementary Row Operations 1. (Replacement) Replace one row by the sum of itself and a

a22 x2

a2n xn
b2
am1x1 am2 x2 amn xn bm
§1.1 Systems of Linear Equations
Confficient matrix and augmented matrix
Coefficient matrix
§1.2 Row Reduction and Echelon Forms
The following matrices are in echelon form:
pivot position
The following matrices are in reduced echelon form:
§1.2 Row Reduction and Echelon Forms
A solution to a system of equations
A system of linear equations has either
1. No solution, or
inconsistent
｝ 2. Exactly one solution, or
3. Infinitely many solutions.
CHAPTER 1 Linear Equations in Linear Algebra
Chapter 1 Linear Equation in Linear Algebra
§ 1.1 Systems of Linear Equations § 1.2 Row Reduction and Echelon Forms § 1.3 Vector Equation § 1.4 The Matrix Equation Ax = b § 1.5 Solution Sets of Linear Systems § 1.7 Linear Independence § 1.8 Introduction to Linear Transformation § 1.9 The Matrix of a Linear Transformation
1.1 Systems of Linear Equations
1. linear equation a1x1 + a2x2+ . . . + anxn = b
Systems of Linear Equations
a11x1 a12 x2 a1n xn b1

a21x1
multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant.
Examples
1. Solving a Linear System