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Linear algebra

Linear algebra. Motivating example: Web start-up. Eigenvector-eigenvalue analysis. Vector space and basis. Linear operators and representations. +. Example: Modeling a freemium cloud data storage business. Free. Premium. +. +. +.

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Linear algebra

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  1. Linear algebra Motivating example: Web start-up Eigenvector-eigenvalue analysis Vector space and basis Linear operators and representations +

  2. Example: Modeling a freemium cloud data storage business Free Premium + + +

  3. Example: Modeling a freemium cloud data storage business Free Premium

  4. Example: Modeling a freemium cloud data storage business Free Premium

  5. Example: Modeling a freemium cloud data storage business Free Premium

  6. Example: Modeling a freemium cloud data storage business Free Premium

  7. Linear algebra Motivating example: Web start-up Eigenvector-eigenvalue analysis Vector space and basis Linear operators and representations +

  8. Vector A vector is an arrow. The position of the head in relation to the tail is expressed in terms of a magnitude and direction.

  9. A set of vectors

  10. A set of vectors

  11. A set of vectors vs. a vector space This scaling (doubling length in this example) ofproduced 2, which belongs to our original set of vectors “Head-to-tail” addition ofand produced a resultant vector not belonging to our original set of vectors A vector space is a set of vectors that is “closed” under scaling and vector addition. Neither scaling nor vector addition produces a result not already included in the “space.”

  12. Basis A vector space is a set of vectors that are “closed” under scaling and vector addition. A set of vectors ,, . . . Linear combination: addition of vectors with scalings Used a set of vectors to prescribe a vector space!

  13. Basis set: Can’t remove any vector without changing space A vector space is a set of vectors that are “closed” under scaling and vector addition. A set of vectors ,, . . . Linear combination: addition of vectors with scalings Basis forV 2-dimensional vector spaceV N W E S

  14. Basis: Coordinate system A vector space is a set of vectors that are “closed” under scaling and vector addition. A set of vectors ,, . . . Linear combination: addition of vectors with scalings

  15. Basis: Coordinate system A vector space is a set of vectors that are “closed” under scaling and vector addition. A set of vectors ,, . . . Linear combination: addition of vectors with scalings

  16. Basis: Coordinate system A vector space is a set of vectors that are “closed” under scaling and vector addition. A set of vectors ,, . . . Linear combination: addition of vectors with scalings

  17. Linear algebra Motivating example: Web start-up Eigenvector-eigenvalue analysis Vector space and basis Linear operators and representations +

  18. Operator Given a vector, an operatoroutputs a vector, possibly scaled and/or rotated A function associates objects from a domain with objects in a codomain, sometimes in terms of elementary arithmetic operations.

  19. Linear operators Scaling Addition

  20. Representing linear operators

  21. Representing linear operators Abstract action on vector Relationship between coefficients Representation in the context of a particular basis “The vector v-prime is represented by the column vector v-prime-sub-1, v-prime-sub-2” “The operator A is represented by the matrix A”

  22. Vector transformation algorithm implies matrix multiplication

  23. Linear algebra Motivating example: Web start-up Eigenvector-eigenvalue analysis Vector space and basis Linear operators and representations +

  24. Example: Modeling a freemium cloud data storage business Free Premium + + +

  25. Example: Modeling a freemium cloud data storage business Free Premium

  26. Example: Modeling a freemium cloud data storage business Free Premium

  27. Example: Modeling a freemium cloud data storage business Free Premium

  28. Example: Modeling a freemium cloud data storage business Free Premium

  29. Example: Modeling a freemium cloud data storage business Free Premium M copies of matrix

  30. Example: Modeling a freemium cloud data storage business Easy-looking-one-dimensional problem Free + Premium M copies of matrix

  31. Example: Modeling a freemium cloud data storage business Check that is consistent in a matrix representation Free Premium M copies of matrix STOP

  32. Example: Modeling a freemium cloud data storage business ] - [ Free Premium M copies of matrix

  33. Example: Modeling a freemium cloud data storage business Free There are 2 possibly special scaling factors. Does each l actually correspond to a special ? Premium M copies of matrix

  34. Example: Modeling a freemium cloud data storage business There are 2 possibly special scaling factors. Does each l actually correspond to a special ? Free Premium M copies of matrix

  35. Example: Modeling a freemium cloud data storage business Free There are 2 special scaling factors; each l corresponds to a special vector . Unless something is hokey, they point in different directions and can serve as a basis. Premium M copies of matrix

  36. Example: Modeling a freemium cloud data storage business Inaugural trials Free Premium M copies of matrix

  37. Example: Modeling a freemium cloud data storage business Inaugural trials Free Premium M copies of matrix

  38. Example: Modeling a freemium cloud data storage business Free Premium M copies of matrix

  39. Example: Modeling a freemium cloud data storage business Free Premium M copies of matrix

  40. Example: Modeling a freemium cloud data storage business Free Premium M copies of matrix

  41. Example: Modeling a freemium cloud data storage business Free Premium , , , = 0.2, 0.2, 0.1, 0.1 M copies of matrix

  42. Example: Modeling a freemium cloud data storage business Free Premium M copies of matrix , , , = 0.2, 0.2, 0.1, 0.1

  43. Example: Modeling a freemium cloud data storage business Eigenvectors Free Eigenvalues Premium , , , = 0.2, 0.2, 0.1, 0.1 M copies of matrix

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