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Understanding the Projection of Vectors: Theorem and Applications in Linear Algebra

This chapter delves into Theorem 5.1.1, which addresses the relationship between two nonzero vectors in R² or R³ using the scalar product. It highlights how to calculate the angle θ between vectors x and y, illustrating the concept of projection. Theorem applications are explored through multiple examples, enhancing comprehension of vector projection and orthogonality. This foundational knowledge is crucial for scholars and practitioners in linear algebra and its applications, setting the stage for more complex topics in vector spaces.

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Understanding the Projection of Vectors: Theorem and Applications in Linear Algebra

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  1. Chapter 5 Section 1

  2. Theorem 5.1.1: If x and y are two nonzero vectors in either R2 or R3 and θ is the angle between them, then xTy=||x||||y||cosθ y-x x y θ

  3. Example

  4. The projection of x onto y x y θ

  5. The projection of x onto y x y θ

  6. The projection of x onto y x y θ u

  7. The projection of x onto y x y θ αu u

  8. The projection of x onto y x x-αu y θ αu u

  9. The projection of x onto y x x-αu y θ αu

  10. The projection of x onto y x x-αu y θ αu

  11. The projection of x onto y x x-αu y θ αu

  12. The projection of x onto y x x-αu y θ αu

  13. Example

  14. Example

  15. Example

  16. Example y=(1,2)T x=(5,2)T

  17. Equations of Planes

  18. Scalar Product: Distance: Angle: Orthogonal when: Extensions to Rn:

  19. Scalar Product: Distance: Angle: Orthogonal when: Extensions to Rn:

  20. Scalar Product: Distance: Angle: Orthogonal when: Extensions to Rn:

  21. Scalar Product: Distance: Angle: Orthogonal when: Extensions to Rn:

  22. Vectors inRn:

  23. Vectors inRn:

  24. Vectors inRn:

  25. Vectors inRn:

  26. Vectors inRn:

  27. Vectors inRn: If x and y are orthogonal….. y x x+y

  28. Search Engines Books: B1: Applied Linear Algebra B2: Elementary Linear Algebra B3: Elementary Linear Algebra with Applications B4: Linear Algebra and its applications B5: Linear Algebra with Applications B6: Matrix Algebra with Applications B7: Matrix Theory Keywords: algebra, application, elementary, linear, matrix, theory

  29. Search Engines Books: B1: Applied Linear Algebra B2: Elementary Linear Algebra B3: Elementary Linear Algebra with Applications B4: Linear Algebra and its applications B5: Linear Algebra with Applications B6: Matrix Algebra with Applications B7: Matrix Theory

  30. Search Engines

  31. Search Engines Words to Search For: Applied Linear Algebra

  32. Search Engines Words to Search For: Applied Linear Algebra

  33. Search Engines

  34. Search Engines B1: Applied Linear Algebra B2: Elementary Linear Algebra B3: Elementary Linear Algebra with Applications B4: Linear Algebra and its applications B5: Linear Algebra with Applications B6: Matrix Algebra with Applications B7: Matrix Theory

  35. Relative Frequency Searches

  36. Relative Frequency Searches

  37. Relative Frequency Searches

  38. Relative Frequency Searches

  39. Relative Frequency Searches Words to Search For: orthogonality spaces vector

  40. Relative Frequency Searches Words to Search For: orthogonality spaces vector Note: Since y5=0.635 is the entry ofythat is closest to 1, the direction of the search vectoruis closest to the direction of q5 and hence document 5 is the one that best matches our search criteria

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