4.3 Linearly Independent Sets; Bases

Slides:

Advertisements

Similar presentations

Vector Spaces A set V is called a vector space over a set K denoted V(K) if is an Abelian group, is a field, and For every element vV and K there exists.

Advertisements

Example: Given a matrix defining a linear mapping Find a basis for the null space and a basis for the range Pamela Leutwyler.

8.2 Kernel And Range.

MAT 2401 Linear Algebra Exam 2 Review

Vector Spaces & Subspaces Kristi Schmit. Definitions A subset W of vector space V is called a subspace of V iff a.The zero vector of V is in W. b.W is.

4 4.3 © 2012 Pearson Education, Inc. Vector Spaces LINEARLY INDEPENDENT SETS; BASES.

THE DIMENSION OF A VECTOR SPACE

Basis of a Vector Space (11/2/05)

Some Important Subspaces (10/29/04) If A is an m by n matrix, then the null space of A is the set of all vectors x in R n which satisfy the homogeneous.

Orthogonality and Least Squares

4 4.6 © 2012 Pearson Education, Inc. Vector Spaces RANK.

Subspaces, Basis, Dimension, Rank

Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ.

Linear Algebra Lecture 25.

4 4.2 © 2012 Pearson Education, Inc. Vector Spaces NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS.

4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transformations 4.3 Linearly Independent Sets; Bases 4.4 Coordinate systems.

Sections 1.8/1.9: Linear Transformations

Section 4.1 Vectors in ℝ n. ℝ n Vectors Vector addition Scalar multiplication.

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.

Chapter 3 Vector Spaces. The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however,

Chapter Content Real Vector Spaces Subspaces Linear Independence

Orthogonality and Least Squares

4 4.6 © 2012 Pearson Education, Inc. Vector Spaces RANK.

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.

Vector Spaces RANK © 2016 Pearson Education, Inc..

4 © 2012 Pearson Education, Inc. Vector Spaces 4.4 COORDINATE SYSTEMS.

LAHW#12 Due December 13, Bases and Dimension 42. –Criticize this argument: We have three vectors u 1 = (1, 3, 2), u 2 = (-2, 4, 5), and u 3.

Class 24: Question 1 Which of the following set of vectors is not an orthogonal set?

5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.

Vector Spaces Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

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.

Linear Transformations

4 4.1 © 2016 Pearson Education, Ltd. Vector Spaces VECTOR SPACES AND SUBSPACES.

4.5: The Dimension of a Vector Space. Theorem 9 If a vector space V has a basis, then any set in V containing more than n vectors must be linearly dependent.

Matrices and Linear Systems Roughly speaking, matrix is a rectangle array We shall discuss existence and uniqueness of solution for a system of linear.

4 4.5 © 2016 Pearson Education, Inc. Vector Spaces THE DIMENSION OF A VECTOR SPACE.

4 4.2 © 2016 Pearson Education, Inc. Vector Spaces NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS.

4 Vector Spaces 4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transformations 4.3 Linearly Independent Sets; Bases 4.4 Coordinate.

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.

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.

4 4.1 © 2012 Pearson Education, Inc. Vector Spaces VECTOR SPACES AND SUBSPACES.

 Matrix Operations  Inverse of a Matrix  Characteristics of Invertible Matrices …

REVIEW Linear Combinations Given vectors and given scalars

Linear Algebra Lecture 26.

Vector Spaces Prepared by Vince Zaccone

VECTOR SPACES AND SUBSPACES

Section 4.1: Vector Spaces and Subspaces

Linear Transformations

Section 1.8: Introduction to Linear Transformations.

Section 4.1: Vector Spaces and Subspaces

Vector Spaces Prepared by Vince Zaccone

Linear Algebra Lecture 22.

VECTOR SPACES AND SUBSPACES

Linear Algebra Lecture 37.

Linear Algebra Lecture 21.

Theorems about LINEAR MAPPINGS.

Mathematics for Signals and Systems

Linear Algebra Lecture 24.

Linear Algebra Lecture 20.

Chapter 4 Vector Spaces 4.1 Vector spaces and Subspaces

Vector Spaces, Subspaces

ABASAHEB KAKADE ARTS COLLEGE BODHEGAON

Vector Spaces RANK © 2012 Pearson Education, Inc..

THE DIMENSION OF A VECTOR SPACE

Null Spaces, Column Spaces, and Linear Transformations

NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS

VECTOR SPACES AND SUBSPACES

Presentation transcript:

4.3 Linearly Independent Sets; Bases 4 Vector Spaces 4.3 Linearly Independent Sets; Bases

REVIEW Definition A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms:

REVIEW Definition A subspace of a vector space V is a subset H of V that satisfies The zero vector of V is in H. H is closed under vector addition. H is closed under multiplication by scalars.

REVIEW Theorem 1 If are in a vector space V, then Span is a subspace of V.

REVIEW Definition The null space of an matrix A, written as Nul A, is the set of all solutions to Ax=0. Theorem 2 The null space of an matrix A is a subspace of .

REVIEW Definition The column space of an matrix A, written as Col A, is the set of all linear combinations of the columns of A. Note: Theorem 3 The column space of an matrix A is a subspace of .

REVIEW Definition: A linear transformation T from a vector space V into a vector space W is a rule assigns to each vector x in V a unique vector T(x) in W, such that (i) T(u+v)=T(u)+T(v) for all u, v in the domain of T: (ii) T(cu)=cT(u) for all u and all scalars c. If T(x)=Ax for some matrix A, Kernel of T = Nul A Range of T = Col A.

4.3 Linearly Independent Sets; Bases Purpose: To study the vectors that span a vector space (or a subspace) as efficiently as possible.

Linear Independence in V is linearly independent has only the trivial solution.

Tips to determine the linear dependence A set is linearly dependent, if it satisfies one of the following: A set has two vectors and one is a multiple of the other. 2. A set has two or more vectors and one of the vectors is a linear combination of the others. 3. A set contains more vectors than the number of entries in each vector. A set contains the zero vector.

Theorem 4 An indexed set of two or more vectors, with , is linearly dependent if and only if some is a linear combination of the preceding vectors, Example:

Definition Let H be a subspace of a vector space V. An indexed set of vectors in V is a basis for H if i) is a linearly independent set, and ii) the subspace spanned by coincides with H; i.e.

Examples: 1. Let A be an invertible matrix. Then the columns of A form a basis for . Why? 2. Let be the columns of the identity matrix I. Then, is called the standard basis for .

4. Let . Determine if is a basis for . 5. Let Determine if is a basis for . 6. Let Determine if is a basis for .

The Spanning Set Theorem Let be a set in V, and let . If one of the vectors in S, say , is a linear combination of the remaining vectors in S, then the set formed from S by removing still spans H. b. If , some subset of S is a basis for H.

Example: Find a basis for Col A, where

Example: Find a basis for Col B, where

Theorem The pivot columns of a matrix A form a basis for Col A.