Lecture 1: Vectors and Vector SpacesAMath 352Mon., Mar. 291 / 12Representation of Vectors0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x-0.200.20.40.60.811.2yFigure: A vector in the plane.IMagnitude:√22+ 12=√5.IDirection: θ = arctan(1/2).Usually write vectors as columns:~v =21,~vT= [2, 1].2 / 12Representation of Vectors0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x-0.200.20.40.60.811.2yFigure: A vector in the plane.IMagnitude:√22+ 12=√5.IDirection: θ = arctan(1/2).Usually write vectors as columns:~v =21,~vT= [2, 1].2 / 12Vector Arithmetic-1 -0.5 0 0.5 1 1.5 200.511.522.533.54Sum of two vectorsvwv+wFigure: Sum of two vectors.~v +~w =21+−13=14.2~v = 221=42.3 / 12n-dimensional Vectors~v =v1v2...vn,~v ∈ Rn, each component vjis a real number.If~w = [w1, w2, . . . , wn]Tand α is a scalar, then~v +~w =v1+ w1v2+ w2...vn+ wn, α~v =αv1αv2...αvn.4 / 12n-dimensional Vectors~v =v1v2...vn,~v ∈ Rn, each component vjis a real number.If~w = [w1, w2, . . . , wn]Tand α is a scalar, then~v +~w =v1+ w1v2+ w2...vn+ wn, α~v =αv1αv2...αvn.4 / 12Vector SpaceA vector space is a collection of objects for which addition andscalar multiplication are defined in such a way that the result ofthese operations is also in the collection.Example. Polynomials of degree at most 2: c0+ c1x + c2x2.(c0+c1x+c2x2)+(d0+d1x+d2x2) = c0+d0+(c1+d1)x+(c2+d2)x2,α(c0+ c1x + c2x2) = αc0+ αc1x + αc2x2.A subset S of a vector space that is also a vector space is called asubspace. Example:S :=all scalar multiples of11is a subspace of R2.5 / 12Vector SpaceA vector space is a collection of objects for which addition andscalar multiplication are defined in such a way that the result ofthese operations is also in the collection.Example. Polynomials of degree at most 2: c0+ c1x + c2x2.(c0+c1x+c2x2)+(d0+d1x+d2x2) = c0+d0+(c1+d1)x+(c2+d2)x2,α(c0+ c1x + c2x2) = αc0+ αc1x + αc2x2.A subset S of a vector space that is also a vector space is called asubspace. Example:S :=all scalar multiples of11is a subspace of R2.5 / 12Linear Independence and DependenceA set of vectors~v1, . . . ,~vmin a vector space V is said to belinearly independent if the only scalars c1, . . . , cmfor whichc1~v1+ . . . + cm~vm=~0 are c1= . . . = cm= 0. Otherwise,~v1, . . . ,~vmare linearly dependent.Examples:100,010,001are linearly independent, sincec1100+ c2010+ c3001=000⇒ c1= c2= c3= 0.6 / 12Linear Independence and Dependence, Cont.123,456,579,are linearly dependent, since1 ·123+ 1 ·456+ (−1) ·579=000.A sum of scalar multiples of vectors c1~v1+ . . . + cm~vmis called alinear combination of~v1, . . . ,~vm.7 / 12Span of a Set of VectorsThe span of~v1, . . . ,~vmis the set of all linear combinations:c1~v1+ . . . + cm~vm.Example:span100,110= {all vectors in R3whose third entry is 0}.If~v1, . . . ,~vmare in a vector space V , then span{~v1, . . . ,~vm} is asubspace of V , since it is closed under addition and scalarmultiplication. It is all of V if every vector~v ∈ V can be writtenas a linear combination of~v1, . . . ,~vm.8 / 12Span of a Set of VectorsThe span of~v1, . . . ,~vmis the set of all linear combinations:c1~v1+ . . . + cm~vm.Example:span100,110= {all vectors in R3whose third entry is 0}.If~v1, . . . ,~vmare in a vector space V , then span{~v1, . . . ,~vm} is asubspace of V , since it is closed under addition and scalarmultiplication. It is all of V if every vector~v ∈ V can be writtenas a linear combination of~v1, . . . ,~vm.8 / 12Basis of a Vector SpaceIf~v1, . . . ,~vmare linearly independent and span V , they are said toform a basis for V .Examples:10,01is a basis for R2.10,11is a basis for R2.11,22is not because they do not span R2and are not lin. indep.10,01,11spans R2but is not a basis since not lin. indep.9 / 12Dimension of a Vector SpaceThere are infinitely many choices for a basis of a vector space V ,but it can be shown that they all have the same number of vectors.This is the dimension of V .Examples: R2has dimension 2. Rnhas dimension n.{polynomials of degree at most 3} has dimension 4; one basis is{1, x, x2, x3}; another is {1, x, 2x2− 1, 4x3− 3x}.10 / 12Coordinates with Respect to a BasisGiven a basis~v1, . . . ,~vmfor V , every~v ∈ V can be writtenuniquely as a linear combination of~v1, . . . ,~vm:~v =Pmj=1cj~vj.c1, . . . , cmare called the coordinates of~v with respect to thebasis~v1, . . . ,~vm.Example: The coordinates of12wrt10,01are 1, 2.The coordinates of12wrt10,11are −1, 2.11 / 12SummaryWe’ve mainly introduced definitions. Maybe not the most excitingthing, but you need to know these definitions in order tocommunicate in linear algebra. Learn these definitions well:1. vector space, subspace,2. linear independence and dependence,3. span of a set of vectors,4. basis of a vector space, dimension of a vector space,coordinates of a vector wrt a basis.12 /