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: reduces 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:

8. 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!

9. A spanning set can be an efficient way to describe a vector space containing infinitely many vectors. SPANS R 2 - but is it the most efficient way to describe R 2 ? Why do we need this? It is a linear combination of (depends on) the other two.

10. A spanning set can be an efficient way to describe a vector space containing infinitely many vectors. SPANS R 2 - but is it the most efficient way to describe R 2 ? Why do we need this? It is a linear combination of (depends on) the other two. This independent set still spans R 2, 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.

11.  =  SPANS R 2 : Given any x and y there exist c 1 and c 2 such that

12.  =  is INDEPENDENT: A linear combination of these vectors produces the zero vector ONLY IF c 1 and c 2 are both zero.

13.  =  is INDEPENDENT and  SPANS R 2 …. Therefore  is a BASIS for R 2. is called the standard basis for R 2  is a nonstandard basis - why do we need nonstandard bases?

14. Consider the points on the ellipse below: Described relative to the standard basis they are solutions to: 8x 2 + 4xy + 5y 2 = 1 Described relative to the  basis they are solutions to: 9x 2 + 4y 2 = 1  basis =

15. There are lots of different ways to write v as a linear combination of the vectors in the set   = v = not a BASIS for R 2 example:

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

17. 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 0000 =

18. 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

20. This is the vector v

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

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

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

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. 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 ? +

30. 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 ? +

31. 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 =0 IF

32. 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 ? =0 IF The rank is less than the number of variables The solution is not unique NO 3 VECTORS CAN NEVER BE INDEPENDENT in a VECTOR SPACE that is SPANNED BY 2 VECTORS

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

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

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

36. An independent set of vectors that does not span V can be “padded” to make a basis for V. theorem: 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 in. That is: Is an independent set.

37. A spanning set that is not independent can be “weeded” to make a basis. theorem: 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.

38. theorem:

39. If the dimension of V is n then the set n Containing n vectors is INDEPENDENT IF AND ONLY IF it SPANS V

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

41. theorem: If the dimension of V is n then the set n Containing n vectors is INDEPENDENT IF AND ONLY IF it SPANS V If S = spans V then S is independent. is independent then S spans V. 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

42. theorem: If the dimension of V is n then the set n Containing n vectors is INDEPENDENT IF AND ONLY IF it SPANS V If S =spans V then S is independent. 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 If S = is independent then S spans V.

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