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Vector Spaces Bases Algebras Beyond Fields: Vector Spaces and Algebras Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras Vector Spaces Bases

  1. Vector Spaces Bases Algebras Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  2. Vector Spaces Bases Algebras Definition. An element of a vector space is also called a vector and an element of F is also called a scalar . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  3. Vector Spaces Bases Algebras Definition. An element of a vector space is also called a vector and an element of F is also called a scalar . We will usually refer to the set X as the vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  4. Vector Spaces Bases Algebras Definition. An element of a vector space is also called a vector and an element of F is also called a scalar . We will usually refer to the set X as the vector space. As is customary for multiplications, the dot is usually omitted. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  5. Vector Spaces Bases Algebras Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  6. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  7. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  8. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  9. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  10. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  11. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  12. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  13. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  14. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  15. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  16. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  17. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  18. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) = f ( x )+ g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  19. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) = f ( x )+ g ( x ) = g ( x )+ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  20. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) = f ( x )+ g ( x ) = g ( x )+ f ( x ) = ( g + f )( x ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  21. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) = f ( x )+ g ( x ) = g ( x )+ f ( x ) = ( g + f )( x ) . The neutral element for addition is the function that is equal to 0 ∈ F for all x ∈ X . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  22. Vector Spaces Bases Algebras Example. Let d ∈ N and let F be a field. The set F d : = � � ( x 1 ,..., x d ) : x i ∈ F with componentwise addition and scalar multiplication is a vector space. Example. Let D be a set and let F be a field. The set F ( D , F ) of all functions f : D → F with addition defined pointwise by ( f + g )( x ) : = f ( x )+ g ( x ) and scalar multiplication defined pointwise by ( α · f )( x ) : = α f ( x ) is a vector space. All properties follow from the corresponding pointwise properties for F . For example, for all x ∈ D we have that ( f + g )( x ) = f ( x )+ g ( x ) = g ( x )+ f ( x ) = ( g + f )( x ) . The neutral element for addition is the function that is equal to 0 ∈ F for all x ∈ X . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  23. Vector Spaces Bases Algebras Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  24. Vector Spaces Bases Algebras Definition. Let X be a vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  25. Vector Spaces Bases Algebras Definition. Let X be a vector space. A subset S ⊆ X \{ 0 } is called linearly independent iff for all finite subsets { x 1 ,..., x n } ⊆ S and all sets of scalars { α 1 ,..., α n } ⊆ F the n ∑ α i x i = 0 implies α 1 = α 2 = ··· = α n = 0 . equation i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  26. Vector Spaces Bases Algebras Definition. Let X be a vector space. A subset S ⊆ X \{ 0 } is called linearly independent iff for all finite subsets { x 1 ,..., x n } ⊆ S and all sets of scalars { α 1 ,..., α n } ⊆ F the n ∑ α i x i = 0 implies α 1 = α 2 = ··· = α n = 0 . equation i = 1 n ∑ α i x i with α i ∈ F and x i ∈ X is also called a linear A sum i = 1 combination of x 1 ,..., x n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  27. Vector Spaces Bases Algebras Definition. Let X be a vector space. A subset S ⊆ X \{ 0 } is called linearly independent iff for all finite subsets { x 1 ,..., x n } ⊆ S and all sets of scalars { α 1 ,..., α n } ⊆ F the n ∑ α i x i = 0 implies α 1 = α 2 = ··· = α n = 0 . equation i = 1 n ∑ α i x i with α i ∈ F and x i ∈ X is also called a linear A sum i = 1 combination of x 1 ,..., x n . Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  28. Vector Spaces Bases Algebras Definition. Let X be a vector space. A subset S ⊆ X \{ 0 } is called linearly independent iff for all finite subsets { x 1 ,..., x n } ⊆ S and all sets of scalars { α 1 ,..., α n } ⊆ F the n ∑ α i x i = 0 implies α 1 = α 2 = ··· = α n = 0 . equation i = 1 n ∑ α i x i with α i ∈ F and x i ∈ X is also called a linear A sum i = 1 combination of x 1 ,..., x n . Definition. A linearly independent set B ⊆ X such that for every x ∈ X there are a finite subset { b 1 ,..., b n } ⊆ B and a set of n ∑ scalars { α 1 ,..., α n } ⊆ F so that x = α i b i is called a base of i = 1 a vector space. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  29. Vector Spaces Bases Algebras Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  30. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  31. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  32. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  33. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  34. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d ∑ α i e i = 0 i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  35. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0 i i = 1 i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  36. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0, which i i = 1 i = 1 for each j simply state that α j = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  37. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0, which i i = 1 i = 1 for each j simply state that α j = 0, as was to be proved. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  38. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0, which i i = 1 i = 1 for each j simply state that α j = 0, as was to be proved. Representation of elements: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  39. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0, which i i = 1 i = 1 for each j simply state that α j = 0, as was to be proved. Representation of elements: For each x = ( x 1 ,..., x d ) ∈ F d we d ∑ have that x = ( x 1 ,..., x d ) = x i e i . i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  40. Vector Spaces Bases Algebras Example. In F d , let e i denote the vector such that the i th component is 1 and all other components are zero. Then { e 1 ,..., e d } is a base of F d . Linear independence: For each i = 1 ,..., d let e ( j ) denote the j th i component of e i . Then for any α 1 ,..., α d the vector equation d d α i e ( j ) ∑ ∑ α i e i = 0 leads to the scalar equations = 0, which i i = 1 i = 1 for each j simply state that α j = 0, as was to be proved. Representation of elements: For each x = ( x 1 ,..., x d ) ∈ F d we d ∑ have that x = ( x 1 ,..., x d ) = x i e i . i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  41. Vector Spaces Bases Algebras Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  42. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  43. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  44. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  45. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  46. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  47. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  48. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . � � ˜ � elements than F , then L ∩ F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  49. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  50. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � ˜ � � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  51. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � ˜ � � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  52. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � ˜ � � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  53. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � ˜ � � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  54. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  55. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. Hence, C has more elements than F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  56. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. Hence, C has more elements than F . Because b �∈ F , we � � �� � ( L \{ b } ) ∪ H ∩ F obtain � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  57. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. Hence, C has more elements than F . Because b �∈ F , we � = � � � � �� � ( L \{ b } ) ∪ H ∩ F � ( L ∩ F ) ∪ H obtain � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  58. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. Hence, C has more elements than F . Because b �∈ F , we � = � � � � �� � � > | L ∩ F | ( L \{ b } ) ∪ H ∩ F � ( L ∩ F ) ∪ H obtain logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  59. Vector Spaces Bases Algebras Theorem. Let X be a vector space with a finite base F. Then every linearly independent subset L of X has at most as many elements as F. Moreover, all bases of X have as many elements as F. Proof. Let F = { f 1 ,..., f n } be a finite base of X . Suppose for a contradiction that there is a linearly independent set L ⊆ X that has more elements than F . WLOG assume that L is such that if ˜ L is another linearly independent subset of X with more � ≤ | L ∩ F | . Let b ∈ L \ F and � � ˜ � elements than F , then L ∩ F consider the linearly independent sets L \{ b } and F . There is a � � subset H ⊆ F \ L so that C : = L \{ b } ∪ H is a base of X (good exercise). But L \{ b } is not a base of X (good exercise), so H � = / 0. Hence, C has more elements than F . Because b �∈ F , we � = � � � � �� � � > | L ∩ F | , a ( L \{ b } ) ∪ H ∩ F � ( L ∩ F ) ∪ H obtain contradiction. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  60. Vector Spaces Bases Algebras Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  61. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  62. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  63. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X , then | B | ≤ | F | . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  64. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X , then | B | ≤ | F | . Therefore B is finite. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  65. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X , then | B | ≤ | F | . Therefore B is finite. With the same argument as above, if L ⊆ X is linearly independent, then | L | ≤ | B | . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  66. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X , then | B | ≤ | F | . Therefore B is finite. With the same argument as above, if L ⊆ X is linearly independent, then | L | ≤ | B | . Because F is linearly independent, we obtain | F | ≤ | B | , and hence | F | = | B | . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  67. Vector Spaces Bases Algebras Proof (concl.). Thus no linearly independent subset of X has more elements than F . Now, if B is another base of X , then | B | ≤ | F | . Therefore B is finite. With the same argument as above, if L ⊆ X is linearly independent, then | L | ≤ | B | . Because F is linearly independent, we obtain | F | ≤ | B | , and hence | F | = | B | . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  68. Vector Spaces Bases Algebras Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  69. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  70. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  71. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  72. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  73. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  74. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  75. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . Example. Let D be a set and let F be a field. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  76. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . Example. Let D be a set and let F be a field. The vector space F ( D , F ) of all functions f : D → F is an algebra with the multiplication operation defined pointwise by ( f · g )( x ) : = f ( x ) · g ( x ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  77. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . Example. Let D be a set and let F be a field. The vector space F ( D , F ) of all functions f : D → F is an algebra with the multiplication operation defined pointwise by ( f · g )( x ) : = f ( x ) · g ( x ) . All properties follow from the corresponding pointwise properties for elements of fields. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  78. Vector Spaces Bases Algebras Definition. Let X be a vector space over the field F and let · : X × X → X be a binary operation. Then ( X , · ) is called an algebra iff 1. The multiplication operation is associative, and 2. Multiplication is left- and right distributive over addition. 3. For all α ∈ F and all x , y ∈ X we have that α ( xy ) = ( α x ) y = x ( α y ) . Example. Let D be a set and let F be a field. The vector space F ( D , F ) of all functions f : D → F is an algebra with the multiplication operation defined pointwise by ( f · g )( x ) : = f ( x ) · g ( x ) . All properties follow from the corresponding pointwise properties for elements of fields. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  79. Vector Spaces Bases Algebras Final Comments on Number Systems logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  80. Vector Spaces Bases Algebras Final Comments on Number Systems 1. An algebra with a few more properties (commutativity, unit element, multiplicative inverses) becomes a field. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

  81. Vector Spaces Bases Algebras Final Comments on Number Systems 1. An algebra with a few more properties (commutativity, unit element, multiplicative inverses) becomes a field. 2. So can we “go beyond C ”? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Beyond Fields: Vector Spaces and Algebras

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