Outline. MA 138 Calculus 2 with Life Science Applications Linear Maps (Section 9.3) Graphical Representation of (Column) Vectors. Addition of Vectors

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1 MA 8 Calculus with Life Science Applications Linear Maps (Section 9) Alberto Corso albertocorso@ukedu Department of Mathematics Uniersit of Kentuck Wednesda, March 8, 07 Outline We mostl focus on matrices, but point out that we can generalize our discussion to arbitrar n n matrices Consider a map of the form a b c d or, in short, A where A is a matri and is a (column) ector Since A is a ector, this map takes a ector and maps it into a ector This enables us to appl A repeatedl: We can compute A(A) A, which is again a ector, and so on We will first look at ectors, then at maps A, and finall at iterates of the map A (ie, A, A, and so on) ma8 ma8 Graphical Representation of (Column) Vectors We assume that is a matri We call a column ector or simpl a ector Since a matri has just two components, we can represent a ector in the plane For instance, to represent the ector in the - plane, we draw an arrow from the origin (0, 0) to the point (, ) Addition of Vectors Because ectors are matrices, we can add ectors using matri addition For instance, 4 This ector sum has a simple geometric representation The sum w is the diagonal in the parallelogram that is formed b the two ectors and w The rule for ector addition is therefore referred to as the parallelogram law w w ma8 ma8

2 Length of Vectors The length of the ector, denoted b, is the distance from the origin (0, 0) to the point (, ) B Pthagoras Theorem we hae length of We define the direction of as the angle α between the positie -ais and the ector The angle α is in the interal 0, π) and satisfies tan α / We thus hae two distinct was of representing ectors in the plane: We can use either the endpoint (, ) or the length and direction (, α) ma8 α Scalar Multiplication of Vectors Multiplication of a ector b a scalar is carried out componentwise If we multipl b, we get This operation corresponds to 4 changing the length of the ector b the factor If we multipl b, then the resulting ector is, which has the same length as the original ector, but points in the opposite direction ma8 Linear Maps (also called Linear Transformations) We start with a graphical approach to stud maps of the form A where A is a matri and is a ector Since A is a ector as well, the map A takes the ector and maps it to the ector A can be thought of as a map from the plane R to the plane R We will discuss simple eamples of maps from R into R defined b A, that take the ector and rotate, stretch, or contract it For an arbitrar matri A, ectors ma be moed in a wa that has no simple geometric interpretation Eample (Reflections) Describe how multiplication b the matrices 0 0 A A 0 0 A 0 0 A ma8 ma8

3 Eample (Contractions or Epansions) Describe how multiplication b the matrices 0 0 A A 0 a 0 0 A A4 0 / a 0 0 b Eample (Shears) Describe how multiplication b the matrices a A A 0 0 A a 0 A4 0 b ma8 ma8 Eample 4 Sir D Arc Wentworth Thompson (Ma, June, 948) was a Scottish biologist, mathematician, and classics scholar He was a pioneer of mathematical biolog Thompson is remembered as the author of the distinctie 97 book On Growth and Form The book led the wa for the scientific eplanation of morphogenesis, the process b which patterns are formed in plants and animals For eample, Thompson illustrated the transformation of Argropelecus offers into Sternopt diaphana b appling a 0 shear mapping ( transection) What is the form of the matri that describes this change? (source: WikipediA) Rotations The following matri rotates a ector in the - plane b an angle α: cos α sin α Rα sin α cos α If α > 0 the rotation is counterclockwise; if α < 0 it is clockwise Properties of Rotations: det(rα) cos α sin α A rotation b an angle α followed b a rotation b an angle β should be equialent to a single rotation b a total angle α β In fact, using the usual trigonometric identities, we hae cos α sin α cos β sin β RαR β sin α cos α sin β cos β cos α cos β sin α sin β cos α sin β sin α cos β sin α cos β cos α sin β sin α sin β cos α cos β cos(α β) sin(α β) R αβ sin(α β) cos(α β) The preious identit shows that the product of rotations is commutatie: RαR β R β Rα ma8 ma8

4 Eample 5 (Rotations) Describe how multiplication b the matrices A B Properties of Linear Maps According to the properties of matri multiplication, the map A satisfies the following conditions: A( w) A Aw, and A(λ) λ(a), where λ is a scalar Because of these two properties, we sa that the map A is linear ma8 ma8 Eample 6 (Problem #, Section 9, p 486) Show b direct calculation that A( w)a Aw and A(λ)λ(A) ( a b ) a b A( w) A Aw c d c d a( ) b( ) c( ) d( ) (a b) (a b ) (c d) (c d ) a b a b c d c d a b a b c d c d Eample 7 Consider A and u and Find Au and A ( ) a b λ a b λ a(λ) b(λ) A(λ) λ(a) c d c d λ λ(a b) c(λ) d(λ) a b λ(c d) λ c d ( ) a b λ c d ma8 ma8

5 Composition of Linear Maps Product of Matrices Consider two linear maps R f R g R gien b the matrices Af and Ag a b α β c d }} γ δ }} Af That is the coordinates are transformed according to the rules a b c d α β γ δ Ag If we compose the two maps we obtain the transformation α(a b) β(c d) (αa βc) (αb βd) γ(a b) δ(c d) (γa δc) (γb δd) whose matri representation corresponds to the product Ag Af of the two matrices αa βc αb βd α β a b γa δc γb δd γ δ c d }}}}}} ma8 Ag f Ag Af