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the course in a nutshell

the course in a nutshell. LINKED TO EXAMPLES. next. V is a vector space of dimension n . S = { v 1 , v 2 , v 3 , . . . , v n } then S is INDEPENDENT if and only if S SPANS V. Return to outline.

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the course in a nutshell

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  1. the course in a nutshell LINKED TO EXAMPLES next

  2. V is a vector space of dimension n. S = { v1 ,v2 ,v3 ,. . . ,vn} then S is INDEPENDENT if and only if S SPANS V. Return to outline

  3. If T is a LINEAR MAPPING then: the dimension of the DOMAIN of T = the dimension of the NULL SPACE of T + the dimension of the RANGE of T Return to outline

  4. Return to outline

  5. next

  6. Reduces to: next

  7. Reduces to: next

  8. next

  9. next

  10. next

  11. return to outline

  12. • = next

  13. • = 6 + -5 + 12 + -4 = 9 return to outline

  14. next

  15. dot product of row 1 of matrix with vector = entry 1 of answer next

  16. dot product of row 2 of matrix with vector = entry 2 of answer return to outline

  17. System of linear equations: Equivalent matrix equation: return to outline

  18. A toy maker manufactures bears and dolls. It takes 4hours and costs $3 to make 1 bear. It takes 2hours and costs $5 to make 1 doll. Find the matrix for T next

  19. A toy maker manufactures bears and dolls. It takes 4hours and costs $3 to make 1 bear. It takes 2hours and costs $5 to make 1 doll. Find the matrix for T return to outline

  20. next

  21. dot product of row 1 of A with column 1 of B = entry in row 1 column 1 of AB next

  22. dot product of row 1 of A with column 2 of B = entry in row 1 column 2 of AB next

  23. dot product of row 2 of A with column 1 of B = entry in row 2 column 1 of AB next

  24. dot product of row 2 of A with column 2 of B = entry in row 2 column 2 of AB return to outline

  25. Reduces to next

  26. A Reduces to return to outline A-1

  27. To find eigenvalues for A, solve for : next

  28. To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 next

  29. To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A next

  30. To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A 2I - A next

  31. To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A 2I - A an eigenvector belonging to 2 is any nonzero multiple of next

  32. To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 eigenvectors are: next

  33. The eigenvalues are 2 and 4 eigenvectors are: A is similar to the diagonal matrix B next

  34. The eigenvalues are 2 and 4 eigenvectors are: B = P –1AP = return to outline

  35. next

  36. next

  37. next

  38. next

  39. Return to outline

  40. A is an nn matrix detA  0 iff A is nonsingular (invertible) iff The columns of A are a basis for Rn iff The null space of A contains only the zero vector iff A is the matrix for a 1-1 linear transformation Return to outline

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