How many solutions? Hung-yi Lee. Review No solution YES NO Given a system of linear equations with m equations and n variables Have solution We don’t.

Slides:



Advertisements
Similar presentations
5.4 Basis And Dimension.
Advertisements

Section 4.6 (Rank).
Some useful linear algebra. Linearly independent vectors span(V): span of vector space V is all linear combinations of vectors v i, i.e.
Matrix Operations. Matrix Notation Example Equality of Matrices.
Class 25: Question 1 Which of the following vectors is orthogonal to the row space of A?
Lecture 14 Second-order Circuits (2)
5 5.1 © 2012 Pearson Education, Inc. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.
 A equals B  A + B (Addition)  c A scalar times a matrix  A – B (subtraction) Sec 3.4 Matrix Operations.
Solving System of Equations Using Graphing
4.2 Graphing linear equations Objective: Graph a linear equation using a table of values.
ECON 1150 Matrix Operations Special Matrices
Chapter 2 Simultaneous Linear Equations (cont.)
4 4.2 © 2012 Pearson Education, Inc. Vector Spaces NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS.
Vectors in R n a sequence of n real number An ordered n-tuple: the set of all ordered n-tuple  n-space: R n Notes: (1) An n-tuple can be viewed.
A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:
A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions.
Objective I will identify the number of solutions a linear system has using one of the three methods used for solving linear systems.
January 22 Review questions. Math 307 Spring 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone.
Chapter Content Real Vector Spaces Subspaces Linear Independence
Sec 3.5 Inverses of Matrices Where A is nxn Finding the inverse of A: Seq or row operations.
Lecture 5 Controlled Sources (1) Hung-yi Lee. Textbook Chapter 2.3, 3.2.
Solving Systems Using Elimination
5.5 Row Space, Column Space, and Nullspace
4.6: Rank. Definition: Let A be an mxn matrix. Then each row of A has n entries and can therefore be associated with a vector in The set of all linear.
Solve a system of linear equations By reducing a matrix Pamela Leutwyler.
Chapter 1 Linear Algebra S 2 Systems of Linear Equations.
Class 24: Question 1 Which of the following set of vectors is not an orthogonal set?
Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A set of equations is called a system of equations. The solution.
Systems of Linear Equations in Vector Form Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.
Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables A ___________________________ x, y, and z is an equation of the.
1.3 Solutions of Linear Systems
Arab Open University Faculty of Computer Studies M132: Linear Algebra
5.4 Third Order Determinants and Cramer’s Rule. Third Order Determinants To solve a linear system in three variables, we can use third order determinants.
GUIDED PRACTICE for Example – – 2 12 – 4 – 6 A = Use a graphing calculator to find the inverse of the matrix A. Check the result by showing.
1.7 Linear Independence. in R n is said to be linearly independent if has only the trivial solution. in R n is said to be linearly dependent if there.
Ch 6 Vector Spaces. Vector Space Axioms X,Y,Z elements of  and α, β elements of  Def of vector addition Def of multiplication of scalar and vector These.
Section 1.7 Linear Independence and Nonsingular Matrices
5 5.1 © 2016 Pearson Education, Ltd. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.
Chapter 5 Chapter Content 1. Real Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Row Space, Column Space, and Nullspace 6.
1 1.4 Linear Equations in Linear Algebra THE MATRIX EQUATION © 2016 Pearson Education, Ltd.
Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces Let V be a set of elements with vector addition and multiplication by scalar is a vector space if these.
Matrices, Vectors, Determinants.
1 SYSTEM OF LINEAR EQUATIONS BASE OF VECTOR SPACE.
Vectors, Matrices and their Products Hung-yi Lee.
 Matrix Operations  Inverse of a Matrix  Characteristics of Invertible Matrices …
REVIEW Linear Combinations Given vectors and given scalars
1.4 The Matrix Equation Ax = b
Inverse of a Matrix Hung-yi Lee Textbook: Chapter 2.3, 2.4 Outline:
Linear independence and matrix rank
Chapter 2 Simultaneous Linear Equations (cont.)
Linear Transformations
Section 1.8: Introduction to Linear Transformations.
4.6: Rank.
Linear Algebra Lecture 37.
Linear Algebra Lecture 3.
What can we know from RREF?
Linear Algebra Lecture 21.
Elementary Linear Algebra
7.5 Solutions of Linear Systems:
Elementary Linear Algebra
Maths for Signals and Systems Linear Algebra in Engineering Lectures 4-5, Tuesday 18th October 2016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN.
Linear Algebra Lecture 6.
Linear Algebra Lecture 7.
How many solutions? Hung-yi Lee New Question:
Section 2.3 Systems of Linear Equations:
Variables and Equations
LINEAR INDEPENDENCE Definition: An indexed set of vectors {v1, …, vp} in is said to be linearly independent if the vector equation has only the trivial.
Null Spaces, Column Spaces, and Linear Transformations
NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS
CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF
Presentation transcript:

How many solutions? Hung-yi Lee

Review No solution YES NO Given a system of linear equations with m equations and n variables Have solution We don’t know how many solutions

Today No solution YES Rank A = n Nullity A = 0 Unique solution Rank A < n Nullity A > 0 Infinite solution NO Given a system of linear equations with m equations and n variables Other cases?

How many solutions? Dependent and Independent

Dependent or Independent? Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

Dependent and Independent Dependent or Independent? Any set contains zero vector would be linear dependent Zero vector is the linear combination of any other vectors Flaw of the definition: How about a set with only one vector? Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

How to check? Given a vector set, {a 1, a 2, , a n }, there exists scalars x 1, x 2, , x n, that are not all zero, such that x 1 a 1 + x 2 a 2 +  + x n a n = 0. Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

Another Definition How about the vector with only one element?

Intuition Intuitive link between dependence and the number of solutions Infinite Solution Dependent: Once we have solution, we have infinite. Dependent: Once we have solution, we have infinite.

Homogeneous Equations Homogeneous linear equations infinite

Homogeneous Equations Columns of A are dependent → If Ax=b have solution, it will have Infinite Solutions If Ax=b have Infinite solutions → Columns of A are dependent Non-zero

How many solutions? Rank and Nullity

Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank

Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank

Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank If A is a mxn matrix: Rank A = n Nullity A = 0 Columns of A are independent

How many solutions? Concluding Remarks

Conclusion No solution YES Rank A = n Nullity A = 0 Unique solution Rank A < n Nullity A > 0 Infinite solution NO

Conclusion Rank A = n Nullity A = 0 YES NO YESNO No solution Unique solution YESNO No solution Infinite solution

Question True or False If the columns of A are linear independent, then Ax=b has unique solution. If the columns of A are linear independent, then Ax=b has at most one solution. If the columns of A are linear dependent, then Ax=b has infinite solution. If the columns of A are linear independent, then Ax=0 (homogeneous equation) has unique solution. If the columns of A are linear dependent, then Ax=0 (homogeneous equation) has infinite solution.

Acknowledgement 感謝 姚鈞嚴 同學發現投影片上的錯誤