SPANNING.

Slides:

Advertisements

Similar presentations

3.2 Bases and Linear Independence

Advertisements

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.

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

6.3 Vectors in the Plane Many quantities in geometry and physics, such as area, time, and temperature, can be represented by a single real number. Other.

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.

5.3 Linear Independence.

THE DIMENSION OF A VECTOR SPACE

Vector Spaces (10/27/04) The spaces R n which we have been studying are examples of mathematical objects which have addition and scalar multiplication.

Vectors and Vector Equations (9/14/05) A vector (for us, for now) is a list of real numbers, usually written vertically as a column. Geometrically, it’s.

 A equals B  A + B (Addition)  c A scalar times a matrix  A – B (subtraction) Sec 3.4 Matrix Operations.

EXAMPLE 3 Identify properties of real numbers

Linear Algebra – Linear Equations

Elementary Linear Algebra Howard Anton Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 1.

Review of basic concepts and facts in linear algebra Matrix HITSZ Instructor: Zijun Luo Fall 2012.

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.

4.8 Rank Rank enables one to relate matrices to vectors, and vice versa. Definition Let A be an m  n matrix. The rows of A may be viewed as row vectors.

VECTORS (Ch. 12) Vectors in the plane Definition: A vector v in the Cartesian plane is an ordered pair of real numbers:  a,b . We write v =  a,b  and.

Class 26: Question 1 1.An orthogonal basis for A 2.An orthogonal basis for the column space of A 3.An orthogonal basis for the row space of A 4.An orthogonal.

A rule that combines two vectors to produce a scalar.

1.3 Solutions of Linear Systems

4.3 Linearly Independent Sets; Bases

Arab Open University Faculty of Computer Studies M132: Linear Algebra

4.1 Introduction to Linear Spaces (a.k.a. Vector Spaces)

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.

4.8 Rank Rank enables one to relate matrices to vectors, and vice versa. Definition Let A be an m  n matrix. The rows of A may be viewed as row vectors.

Vectors in the Plane. Quick Review Quick Review Solutions.

Chapter 4 Vector Spaces Linear Algebra. Ch04_2 Definition 1: ……………………………………………………………………. The elements in R n called …………. 4.1 The vector Space R n Addition.

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

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.

(2 x 1) x 4 = 2 x (1 x 4) Associative Property of Multiplication 1.

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.

Two sets:VECTORS and SCALARS four operations: A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE.

1 1.3 © 2016 Pearson Education, Ltd. Linear Equations in Linear Algebra VECTOR EQUATIONS.

REVIEW Linear Combinations Given vectors and given scalars

Does the set S SPAN R3 ?.

Linear Transformations

Linear Transformations

VECTOR SPACES AND SUBSPACES

Systems of Linear Equations

Linear Transformations

Vectors, Linear Combinations and Linear Independence

Linear Algebra Lecture 22.

VECTOR SPACES AND SUBSPACES

Linear Algebra Chapter 4 Vector Spaces.

1.3 Vector Equations.

State Space Method.

Mathematics for Signals and Systems

Elementary Linear Algebra

Linear Algebra Lecture 5.

Linear Algebra Lecture 20.

Linear Transformations

Vectors and Dot Products

RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE, HADAPSAR, PUNE

Vector Spaces, Subspaces

6.3 Vectors in the Plane Ref. p. 424.

Vectors in the Plane.

General Vector Spaces I

Linear Vector Space and Matrix Mechanics

THE DIMENSION OF A VECTOR SPACE

NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS

Linear Equations in Linear Algebra

VECTOR SPACES AND SUBSPACES

CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF

Presentation transcript:

SPANNING

3 6 = 3 1 2 3 6 is a multiple of 1 2 11 8 = 1 2 4 3 + 2 1 2 4 is a linear combination of 11 8 and

and Is a linear combination of

B is a linear combination of A and C VECTORS SCALARS The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 2 A 1 C = B B is a linear combination of A and C

-1 -1 -1 Reduced echelon form

Every solution to this homogenious system is a linear combination of the vectors:

SPAN V. The system has infinitely many solutions that form a vector space, V. It would not be humanly possible to list all of these solutions. Fortunately, the entire vector space is determined by just two vectors. We say that these two vectors SPAN V. Every solution to this homogenious system is a linear combination of the vectors:

definition: A set of vectors S is said to span a vector space V if every vector in V is a linear combination of the vectors in S.

and SPAN the plane ( ie: R2 ) . . . . . . . . . . . .

and SPAN the plane ( ie: R2 ) Every point is a linear combination of these two vectors. . . . . . . . . . . . . For example: = 5 + 3

and SPAN the plane ( ie: R2 ) Every point is a linear combination of these two vectors. . . . . . . . . . . . . For example: = 2 + 3

. . . . . . . . . . . .