Lecture 1 Vectors and Vector Spaces AMath 352 Mon Mar 29 1 12 Usually write vectors as columns cid 126 v cid 20 2 cid 21 1 cid 126 v T 2 1 Representation of Vectors Figure A vector in the plane cid 73 Magnitude 5 cid 73 Direction arctan 1 2 22 12 2 12 0 20 40 60 811 21 41 61 82x 0 200 20 40 60 811 2y Representation of Vectors Figure A vector in the plane cid 73 Magnitude 5 cid 73 Direction arctan 1 2 Usually write vectors as columns 22 12 cid 126 v cid 21 cid 20 2 1 cid 126 v T 2 1 2 12 0 20 40 60 811 21 41 61 82x 0 200 20 40 60 811 2y Vector Arithmetic Figure Sum of two vectors cid 126 v cid 126 w cid 20 2 1 cid 21 cid 20 1 3 cid 21 cid 20 1 4 cid 21 2 cid 126 v 2 cid 20 2 1 cid 21 cid 20 4 2 cid 21 3 12 1 0 500 511 5200 511 522 533 54Sum of two vectorsvwv w If cid 126 w w1 w2 wn T and is a scalar then cid 126 v cid 126 w cid 126 v v1 w1 v2 w2 vn wn v1 v2 vn n dimensional Vectors v1 v2 vn cid 126 v cid 126 v Rn each component vj is a real number 4 12 n dimensional Vectors v1 v2 vn cid 126 v cid 126 v Rn each component vj is a real number If cid 126 w w1 w2 wn T and is a scalar then cid 126 v cid 126 w cid 126 v v1 w1 v2 w2 vn wn v1 v2 vn 4 12 A subset S of a vector space that is also a vector space is called a subspace Example cid 26 S all scalar multiples of cid 20 1 cid 21 cid 27 1 is a subspace of R2 Vector Space A vector space is a collection of objects for which addition and scalar multiplication are de ned in such a way that the result of these operations is also in the collection Example Polynomials of degree at most 2 c0 c1x c2x 2 c0 c1x c2x 2 d0 d1x d2x 2 c0 d0 c1 d1 x c2 d2 x 2 c0 c1x c2x 2 c0 c1x c2x 2 5 12 Vector Space A vector space is a collection of objects for which addition and scalar multiplication are de ned in such a way that the result of these operations is also in the collection Example Polynomials of degree at most 2 c0 c1x c2x 2 c0 c1x c2x 2 d0 d1x d2x 2 c0 d0 c1 d1 x c2 d2 x 2 c0 c1x c2x 2 c0 c1x c2x 2 A subset S of a vector space that is also a vector space is called a subspace Example cid 26 S all scalar multiples of cid 21 cid 27 cid 20 1 1 is a subspace of R2 5 12 Linear Independence and Dependence A set of vectors cid 126 v1 cid 126 vm in a vector space V is said to be linearly independent if the only scalars c1 cm for which c1 cid 126 v1 cm cid 126 vm cid 126 0 are c1 cm 0 Otherwise cid 126 v1 cid 126 vm are linearly dependent Examples 1 0 0 0 1 0 0 0 1 are linearly independent since 0 1 0 0 0 1 0 0 0 1 0 0 c1 c2 c3 c1 c2 c3 0 6 12 Linear Independence and Dependence Cont 1 2 3 4 5 6 5 7 9 are linearly dependent since 1 1 2 3 1 1 4 5 6 5 7 9 0 0 0 A sum of scalar multiples of vectors c1 cid 126 v1 cm cid 126 vm is called a linear combination of cid 126 v1 cid 126 vm 7 12 If cid 126 v1 cid 126 vm are in a vector space V then span cid 126 v1 cid 126 vm is a subspace of V since it is closed under addition and scalar multiplication It is all of V if every vector cid 126 v V can be written as a linear combination of cid 126 v1 cid 126 vm Span of a Set of Vectors The span of cid 126 v1 cid 126 vm is the set of all linear combinations c1 cid 126 v1 cm cid 126 vm Example span 1 0 0 1 1 0 all vectors in R3 whose third entry is 0 8 12 Span of a Set of Vectors The span of cid 126 v1 cid 126 vm is the set of all linear combinations c1 cid 126 v1 cm cid 126 vm Example span 1 0 0 1 1 0 all vectors in R3 whose third entry is 0 If cid 126 v1 cid 126 vm are in a vector space V then span cid 126 v1 cid 126 vm is a subspace of V since it is closed under addition and scalar multiplication It is all of V if every vector cid 126 v V can be written as a linear combination of cid 126 v1 cid 126 vm 8 12 Basis of a Vector Space If cid 126 v1 cid 126 vm are linearly independent and span V they are said to form a basis for V Examples cid 26 cid 20 1 0 cid 26 cid 20 1 0 cid 21 cid 21 cid 20 0 1 cid 20 1 1 cid 21 cid 27 cid 21 cid 27 is a basis for R2 is a basis for R2 cid 21 cid 27 cid 26 cid 20 1 1 cid 26 cid 20 1 0 cid 21 cid 21 cid 20 2 2 cid 20 0 1 cid 21 cid 27 cid 21 cid 20 1 1 is not because they do not span R2 and are not lin indep spans R2 but is not a basis since not lin indep 9 12 Dimension of a Vector Space There are in nitely 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 R2 has dimension 2 Rn has dimension n polynomials of degree at most 3 has dimension 4 one basis is 1 x x 2 x 3 another is 1 x 2x 2 1 4x 3 3x 10 12 Coordinates with Respect to a Basis Given a basis cid 126 v1 cid 126 vm for V every cid 126 v V can be written uniquely as a linear combination of cid 126 v1 cid 126 vm cid 126 v cid 80 m j 1 cj cid 126 vj c1 cm are called the coordinates of cid 126 v with respect to the basis cid 126 v1 cid 126 vm Example The coordinates of are 1 2 cid 20 1 2 cid 20 1 0 cid 21 cid 21 cid 20 1 0 …