中山大学~线性代数期末总复习

if and only if an echelon form of the augmented matrix has no
row of the form
§1.3 Vector Equations

Algebraic Properties of
For all u, v, w in and all scalars c and d:
§1.1 Systems of Linear Equations

Confficient matrix and augmented matrix
Coefficient matrix augmented matrix
a11 a 21 am1
a12 a22 am 2
a1n a2 n amn
§1.4 The Matrix Equation Ax = b

1.Definition
If A is an m×n matrix, with column a1,…,an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of
inconsistent
consistent
｝ 3. Infinitely many solutions.
§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
where –u denotes (-1)u
§1.3 Vector Equations

Subset of - Span {v1,…,vp} is collection of all vectors that can be written in the form
with c1,…,cp scalars.
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
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?
§1.2 Row Reduction and Echelon Forms

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
Step5 Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot.
§1.2 Row Reduction and Echelon Forms

Solution of Linear Systems (Using Row Reduction)
1.1 Systems of Linear Equations

1. linear equation
a1x1 + a2x2+ . . . + anxn = b Systems of Linear Equations
a11 x1 a12 x2 a1n xn b1 a x a x a x b 21 1 22 2 2n n 2 am1 x1 am 2 x2 amn xn bm
Chapter 6 Orthogonality and Least Squares
Chapter 7 Symmetric Matrices and Quadratic Forms
CHAPTER 1 Linear Equations in Linear Algebra
Chapter 1 Linear Equation in Linear Algebra
Ax = b
has the same solution set as the vector equation
which, in turn, has the same solution set as the system of linear equation whose augmented matrix is

The following matrices are in echelon form:
pivot position

The following matrices are in reduced echelon form:
§1.2 Row Reduction and Echelon Forms

Theorem 1 Uniqueness of the Reduced Echelon Form
a11 a 21 am1
a12 a22
a1n a2 n
am 2 amn
b1 b2 bm
§1.1 Systems of Linear Equations

A solution to a system of equations

A system of linear equations has either 1. No solution, or 2. Exactly one solution, or
§1.4 The Matrix Equation Ax = b

2. Existence of Solutions
The equation Ax=b has a solution if and only if b is a linear combination of columns of A.
Example. Is the equation Ax=b consistent for all possible b1,b2,b3?
statements or they are all false.
a. For each b in Rm, the equation Ax = b has a solution. b. Each b in Rm is a linear combination of the columns of A. c. The columns of A span Rm. d. A has a pivot position in every row.
eg. Find the general solution of the following linear system
Solution:
§1.2 Row Reduction and Echelon Forms
The associated system now is
The general solution is:
the columns of A using the corresponding entries in x as weights;
that is:
§1.4 The Matrix Equation Ax = b

Theorem 3
If A is an m×n matrix, with column a1,…,an, and if b is in Rm, the matrix equation
§1.4 The Matrix Equation Ax=b

Solution Row reduce the augmented matrix for Ax=b:

∵ ∴
=
≠ 0 (for some choices of b)
The equation Ax=b is not consistent for every b.
§ 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
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
Each matrix is row equivalent to one and only one reduced echelon matrix.
§1.2 Row Reduction and Echelon Forms
Βιβλιοθήκη Baidu
The Row Reduction Algorithm
Step1 Begin with the leftmost nonzero column. Step2 Select a nonzero entry in the pivot column as a pivot.
§1.4 The Matrix Equation Ax=b

3. Computation of Ax
Example . Compute Ax, where
§1.2 Row Reduction and Echelon Forms

Theorem 2 Existence and Uniqueness Theorem
A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column– that is ,
§1.4 The Matrix Equation Ax=b

Theorem 4
Let A be an m×n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true
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.