MTH 215: Introduction To Linear Algebra - Chapter 4

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessMTH 215: Introduction to Linear AlgebraChapter 4Jonathan A. Chávez Casillas11University of Rhode IslandDepartment of MathematicsFebruary 25, 2019Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt Process1Vectors in RnAlgebra in RnLength of a VectorUnit VectorsThe Dot Product2Linear Independence, Spanning Sets and BasisDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and Basis3Row Space, Column Space and the Null Space of a MatrixThe Row and Column SpacesThe Null SpaceThe Image of a MatrixThe Rank-Nullity Theorem4Orthogonality and the Gram Schmidt ProcessOrthogonal Sets and MatricesOrthogonality and IndependenceGram-Schmidt ProcessJonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductWhat is Rn ?Notation and TerminologyR denotes the set of real numbers.R2 denotes the set of all column vectors with two entries.R3 denotes the set of all column vectors with three entries.In general, Rn denotes the set of all column vectors with n entries.Scalar quantities versus vector quantitiesA scalar quantity has only magnitude; e.g. time, temperature.A vector quantity has both magnitude and direction; e.g. displacement, force, wind velocity.Whereas two scalar quantities are equal if they are represented by the same value, two vector quantities areequal if and only if they have the same magnitude and direction.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductR2 and R3Vectors in R2 and R3 have convenient geometric representations as position vectors of points in the2-dimensional (Cartesian) plane and in 3-dimensional space, respectively.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductNotationIf P is a point in Rn with coordinates (p1 , p2 , ., pn ) we denote this by P (p1 , p2 , ., pn ).If P (p1 , p2 , . . . , pn ) is a point in Rn , then p1p2 # P . pnis often used to denote the position vector of the point.Instead of using a capital letter to denote the vector (as we generally do with matrices), we emphasizethe importance of the geometry and the direction with an arrow over the name of the vector.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductNotation and Terminology# The notation P emphasizes that this vector goes from the origin 0 to the point P. We can also use# lower case letters for names of vectors. In this case, we write P # p.Any vector x1x2 n# x . in Rxnis associated with the point (x1 , x2 , . . . , xn ). Please notice that in some context # x can be a rowvector or a column vector.Often, there is no distinction made between the vector # x and the point (x1 , x2 , . . . , xn ), and we say x1 x2 n that both (x1 , x2 , . . . , xn ) Rn and # x . R .xnJonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductAlgebra in RnAddition in RnSince vectors in Rn are n 1 matrices, addition in Rn is precisely matrix addition using column or rowmatrices, i.e.,If # u and # v are in Rn , then # u # v is obtained by adding together corresponding entries of the vectors.# The zero vector in Rn is the n 1 zero matrix, and is denoted 0 .ExampleLet # u "123#and # v "456#. Then,# u # v "123#Jonathan Chávez" 456#" 579#

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductProperties of Vector Addition# be vectors in Rn . Then the following properties hold.Let # u , # v , and w1234# u # v # v # u# # # # # )(u v ) wu ( # v w# # u 0 # u# # u ( # u) 0(vector addition is commutative).(vector addition is associative).(existence of an additive identity).(existence of an additive inverse).Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductScalar MultiplicationSince vectors in Rn are n 1 matrices, scalar multiplication in Rn is precisely matrix scalar multiplicationusing column matrices, i.e., If # u is a vector in Rn and k R is a scalar, then k # u is obtained by multiplying# every entry of u by k.ExampleLet # u "123#and k 4. Then,k # u 4"Jonathan Chávez123#" 4812#

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductProperties of Scalar MultiplicationLet # u , # v Rn be vectors and k, p R be scalars. Then the following properties hold.1234k( # u # v ) k # u k # v(scalar multiplication distributes over vector addition).(k p) # u k # u p # u(addition distributes over scalar multiplication).k(p # u ) (kp) # u# 1 u # u(scalar multiplication is associative).(existence of a multiplicative identity).Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductLength of a Vector, R2If # x x1 R2 , then the length of the vector # x is the distance from the origin 0 to the pointx2X (x1 , x2 ) given by d(0, X ).The length of # x , denoted # x , is given by:d(0, X ) # x Jonathan Chávez p# x12 x22x T # x

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductLength of a Vector, Rn x1x2 n This extends clearly to # x . R .xnThe length of # x is the distance from the origin 0 to the point X (x1 , x2 , . . . , xn ) given by d(0, X ).d(0, X ) # x p# x T # x x 2 x 2 . . . xn2 .12Please notice that if we define # x [x1 , x2 , . . . , xn ] as a row vector. Then,d(0, X ) # x p# x # xT x12 x22 . . . xn2 .Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductUnit VectorsDefinitionA unit vector is a vector of length one.Example"100# ",010# ",001# , 22 0 , are examples of unit vectors. 22Example# If # v , 0 , then1 # v # v is a unit vector in the same direction as # v.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductExample# v " 132#is not a unit vector, since # v 14. However, 1# u # v 14 1 14 314 214 is a unit vector in the same direction as # v , i.e.,11 # u # v 14 1.1414Example# are nonzero that haveIf # v and wthe same direction, then # v opposite directions, then # v # v # w # ;w # v # w # .wJonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductDefinitionIf u1v1u2 v # # 2 u . , and v . unvnare in Rn , then the dot product # u q# v is as the 1 1 matrix v1v2 # u T # v [u1 , u2 , . . . , un ] . u1 v1 u2 v2 · · · un vnvnwhich is treated as a scalar given by u1 v1 u2 v2 · · · un vnPlease notice that this definition can be adapted if # x and # y are regarded as row vectors. The only changeis that # x q# y # x # y T.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessAlgebra in RnLength of a VectorUnit VectorsThe Dot ProductThe Dot ProductProblemFind # u q# v for # u [1, 2, 0, 1]T , # v [0, 1, 2, 3]T .Solution# u q# v (1)(0) (2)(1) (0)(2) ( 1)(3) 0 2 0 3 1Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and BasisTable of Contents1Vectors in RnAlgebra in RnLength of a VectorUnit VectorsThe Dot Product2Linear Independence, Spanning Sets and BasisDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and Basis3Row Space, Column Space and the Null Space of a MatrixThe Row and Column SpacesThe Null SpaceThe Image of a MatrixThe Rank-Nullity Theorem4Orthogonality and the Gram Schmidt ProcessOrthogonal Sets and MatricesOrthogonality and IndependenceGram-Schmidt ProcessJonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and BasisDefinition# Rn is said to be a linear combination of the vectorsLet # v 1 , # v 1 , . . . , # v k be k vectors on Rn . A vector w# v 1 , # v 1 , . . . , # v k if there exist constants a1 , a2 , . . . , ak (called coefficients) such that# wkXai # v i a1 # v 1 a2 # v 2 . . . ak # vki 1DefinitionLet S { # v 1 , # v 1 , . . . , # v k } be a set of k vectors on Rn . That is, S Rn .The span of S, written as span(S) is the set of all linear combinations of the elements of S. That is,span(S) ( kX)# ai v i such that a1 , a2 . . . ak Ri 1Please notice that for creating the span, we consider ALL possible combinations of the coefficientsa1 , a2 , . . . , ak .Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and BasisDefinitionA set of non-zero vectors { # u 1 , · · · , # u k } in Rn is said to be linearly independent if wheneverkX# ai # ui 0i 1it follows that each ai 0. A set that is not linearly independent is called linearly dependent.We can rewrite the definition of linear independence as follows:# A set of non-zero vectors { # u 1 , · · · , # u k } in Rn is said to be linearly independent if whenever 0 is a linearcombination of them, the coefficients of the linear combination are all 0.Jonathan Chávez

Vectors in RnLinear Independence, Spanning Sets and BasisRow Space, Column Space and the Null Space of a MatrixOrthogonality and the Gram Schmidt ProcessDefinitions and examplesUsing matrices to determine linear independence and spanningSubspaces and BasisA Linearly Dependent SetProblemConsider the vectors # u "012#, # v "023## ,w"041## } linearly independent?. Is the set { # u , # v,wSolution# as a linear combination of # Notice that we can write wu , # v as follows:"041#" ( 10)012#" (7)023## is in span{ # Hence, wu , # v }. By the definition, this set is not linearly independent (it is linearly dependent).Jonathan Chávez