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BASIS

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BASIS

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  1. BASIS

  2. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to: reduces to

  3. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to: reduces to

  4. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to: reduces to

  5. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to: reduces to

  6. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to:

  7. A vector space containing infinitely many vectors can be efficiently described by listing a set of vectors that SPAN the space. eg: describe the solutions to: These two vectors SPAN the set of solutions. Each of the infinitely many solutions is a linear combination of these two vectors!

  8. Why do we need this? It is a linear combination of (depends on) the other two. A spanning set can be an efficient way to describe a vector space containing infinitely many vectors. SPANSR2- but is it the most efficient way to describe R2 ?

  9. A spanning set can be an efficient way to describe a vector space containing infinitely many vectors. SPANSR2- but is it the most efficient way to describe R2 ? Why do we need this? It is a linear combination of (depends on) the other two. This independent set still spans R2 , and is a more efficient way to describe the vector space! definition: An INDEPENDENT set of vectors that SPANS a vector space V is called a BASIS for V.

  10.  =  SPANS R2 : Given anyx and y there exist c1and c2 such that

  11.  =  is INDEPENDENT: A linear combination of these vectors produces the zero vector ONLY IF c1and c2are both zero.

  12.  =  is INDEPENDENT and  SPANS R2 …. Therefore  is a BASIS for R2. is called the standard basis for R2  is a nonstandard basis - why do we need nonstandard bases?

  13.  basis = Consider the points on the ellipse below: Described relative to the  basis they are solutions to: 9x2 + 4y2 = 1 Described relative to the standard basisthey are solutions to: 8x2 + 4xy + 5y2 = 1

  14. example: not a BASIS for R2  = v = There are lots of different ways to write v as a linear combination of the vectors in the set 

  15. theorem: If  = is a BASIS for a vector space V, then for every vector in V there are unique scalars = Such that: the c’sexist because spans V they are unique because  is independent

  16. 0 0 0 0 theorem: If  = is a BASIS for a vector space V, then for every vector in V there are unique scalars = Such that: 0 = ONLY IF

  17. theorem: If  = is a BASIS for a vector space V, then for every vector in V there are unique scalars = Such that: the coordinates of relative to the  basis

  18. Coordinates

  19. This is the vector v

  20. Relative to the standard basis the coordinates of v are 5  = 1

  21. Relative to the basis the coordinates of v are 3 2

  22. coordinates relative standard basis coordinates relative  basis v = =

  23. Theorems about bases

  24. example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ?

  25. example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ?

  26. example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ?

  27. + example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ?

  28. + example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ?

  29. =0 =0 =0 =0 example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ? IF + IF

  30. =0 =0 The solution is not unique example: Suppose V is a vector space that is SPANNED by the two vectors Is it possible that this set of three vectors is INDEPENDENT ? NO IF 3 VECTORS CAN NEVER BE INDEPENDENT in a VECTOR SPACE that is SPANNED BY 2 VECTORS The rank is less than the number of variables

  31. m k SPANS V If is INDEPENDENT in V and then k m The number of independent vectors in a vector space V can never exceed the number of vectors that span V. theorem:

  32. If is a basis for V theorem: then k= m and k k k m IS INDEPENDENT SPANS m m IS INDEPENDENT SPANS k< m k> m Two different bases for the same vector space will contain the same number of vectors. is a basis for V proof:

  33. k m Two different bases for the same vector space will contain the same number of vectors. If is a basis for V theorem: then k= m and is a basis for V definition: The number of vectors in a basisfor V is called the DIMENSIONof V.

  34. Suppose Is independent but does not span V. Then there is at least one vector in V , call it , such that Cannot be written as a linear combination of the vectors . That is: in Is an independent set. An independent set of vectors that does not span V can be “padded” to make a basis for V. theorem:

  35. Suppose spans V but is not independent. Then there is at least one vector in the set, call it u , such that u is a linear combination of the other vectors in the set. Remove u and the remaining vectors in the set will still span V. A spanning set that is not independent can be “weeded” to make a basis. theorem:

  36. theorem:

  37. Containing nvectors theorem: If the dimension of V is n then the set n is INDEPENDENT IF AND ONLY IF it SPANS V

  38. Containing nvectors theorem: If the dimension of V is n then the set n is INDEPENDENT IF AND ONLY IF it SPANS V

  39. Containing nvectors If S = spans V and is not independent then one vector can be removed leaving a spanning set containing n-1 vectors. Since dim V = n, there is in V a set of n independent vectors (basis ). This is impossible. You cannot have more independent vectors than spanning vectors theorem: If the dimension of V is n then the set n is INDEPENDENT IF AND ONLY IF it SPANS V If S = If S = spans V is independent then S spans V. then S is independent.

  40. Containing nvectors If S = is independent and does not span V, then a vector can be added to S making a set containing n+1 independent vectors - impossible in a space spanned by n vectors- a basis for V contains n vectors theorem: If the dimension of V is n then the set n is INDEPENDENT IF AND ONLY IF it SPANS V If S = If S = spans V is independent then S spans V. then S is independent.

  41. Containing nvectors theorem: If the dimension of V is n then the set n is INDEPENDENT IF AND ONLY IF it SPANS V