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Chapter 3 Vector Spaces. The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however,

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Presentation on theme: "Chapter 3 Vector Spaces. The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however,"— Presentation transcript:

1 Chapter 3 Vector Spaces

2 The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however, these operations usually obey the same set of algebra rules. Thus a general theory of mathematical systems involving addition and scalar multiplication will have application to many areas in mathematics. Mathematical systems of this form are called vector spaces or linear spaces.

3 1 Definition and Examples Euclidean Vector Spaces R n In general, scalar multiplication and addition in R n are defined by and for anyand any scalar 。

4 The Vector Space R m×n R m×n denote the set of all m × n matrices with real entries.

5 Definition Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique elements x+y that is also in V, and with each element x in V and each scalar, we can associate a unique element x in V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied. Vector Space Axioms

6 A1. x+y=y+x for any x and y in V. A2. (x+y)+z=x+(y+z) for any x, y, z in V. A3. There exists an element 0 in V such that x+0=x for each x ∈ V. A4. For each x ∈ V, there exists an element –x in V such that x+(-x)=0. A5. α(x+y)= αx+ αy for each scalar α and any x and y in V. A6. (α+β)x=αx+βx for any scalars α and β and any x ∈ V. A7. (αβ)x=α(βx) for any scalars α and β and any x ∈ V. A8. 1·x=x for all x ∈ V.

7 The closure properties of the two operations: C1. If x ∈ V and α is a scalar, then αx ∈ V. C2. If x,y ∈ V, then x+y ∈ V. Example Let W={(a,1) a real} with addition and scalar Multiplication defined in the usual way.

8 Example Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by We use the symbol to denote the addition operation for this system avoid confusion with the usual addition x+y of row vectors. Show that S, with the ordinary scalar multiplication and addition operation,is not a vector space. Which of the eight axioms fail to hold?

9 The Vector Space C[a, b] C[a,b] denote the set of all real-valued functions that are defined and continuous on the closed interval [a,b].

10 The Vector Space P n P n denote the set of all polynomials of degree less than n.

11 Theorem 3.1.1 If V is a vector space anf x is any Element of V, then (1) 0x=0. (2) x+y=0 implies that y=-x. (3) (-1)x=-x.

12 2 Subspaces Example Let, S is a subset of R 2. Definition If S is a nonempty subset of a vector space V, and S satisfies the following conditions: (1) whenever for any scalar (2) whenever and then S is said to be a subspace of V.

13 Example 1 、 Let, S is a subspace of R 3. 2 、 Let. If either of the two conditions in the definition fails to hold, then S will not be a subspace. 3 、 Let.The set S forms a subspace of R 2 × 2.

14 The Nullspace of a Matrix Let A be an m × n matrix. Let N(A) denote the set of all solutions to the homogeneous system Ax=0. Thus The subspace N(A) is called the nullspace of A. Example Determine N(A) if

15 The Span of a Set of Vectors Definition Let v 1, v 2, …, v n be vectors in a vector space V. A sum of the form α 1 v 1 + α 2 v 2 + ‥‥ α n v n, where α 1, …, α n are scalars, is called a linear combination of v 1, v 2, …, v n. The set of all linear combinations of v 1, v 2, …, v n is called the span of v 1, v 2, …, v n. The span of v 1, v 2, …, v n will be denoted by Span(v 1, …, v n ).

16 Theorem 3.2.1 If v 1, v 2, …, v n are elements of a vector space V, then Span(v 1, v 2, …, v n ) is a subspace of V. Spanning Set for a Vector Space Definition The set {v 1, v 2, …, v n } is a spanning set for V if and only if every vector in V can be written as a linear combination of v 1, v 2, ‥‥, v n.

17 Example Which of the following are spanning sets for R 3 ?

18 3 Linear Independence Consider the following vectors in R 3 :

19 Conclusion: (1) If v 1, v 2, …, v n span a vector space V and one of these vectors can be written as a linear combination of the other n-1 vectors, then those n-1 vectors span V. (2) Given n vectors v 1, v 2, …, v n, it is possible to write one of the vectors as a linear combination of the other n-1 vectors if and only if there exist scalars c 1, …, c n not all zero such that c 1 v 1 +c 2 v 2 + ‥‥ c n v n =0

20 Definition The vectors v 1, v 2, …, v n in a vector space V are said to be linearly independent if c 1 v 1 +c 2 v 2 + ‥‥ +c n v n =0 implies that all the scalars c 1, …, c n must equal 0.

21 Definition The vectors v 1, v 2, …, v n in a vector space V are said to be linearly dependent if there exist scalars c 1, …, c n not all zero such that c 1 v 1 +c 2 v 2 + ‥‥ +c n v n =0 If there are nontrivial choices of scalars for which the linear combination c 1 v 1 +c 2 v 2 + ‥‥ +c n v n equals the zero vector, then v 1, v 2, …, v n are linearly dependent. If the only way the linear combination c 1 v 1 +c 2 v 2 + ‥‥ +c n v n can equal the zero vector is for all the scalars c 1, …, c n to be 0, then v 1, v 2, …, v n are linearly independent.

22 Theorems and Examples Example Which of the following collections of vectors are linearly independent in R 3 ?

23 Theorem 3.3.1 If x 1, x 2, …, x n be n vectors in R n and let X=(x 1, …, x n ). The vectors x 1,x 2, …, x n will be linearly dependent if and only if X is singular. Example Determine whether the vectors (4, 2, 3) T, (2, 3, 1) T, and (2, -5, 3) T are linearly dependent.

24 Example Given determine if the vectors are linearly independent.

25 Theorem 3.3.2 If v 1, v 2, …, v n be vectors in a vector space V. A vector v in Span(v 1, …, v n ) can be written uniquely as a linear combination of v 1,v 2, …, v n if and only if v 1,v 2, …, v n are linearly independent.

26 4 Basis and Dimension Definition The vectors v 1, v 2, …, v n form a basis for a vector space V if and only if (1) v 1, …, v n are linearly independent. (2) v 1, …, v n span V.

27 Example The standard basis for R 3 is {e 1, e 2, e 3 };however, there are many bases that we could choose for R 3. Example In R 2×2, consider the set {E 11, E 12, E 21, E 22 }, where

28 Theorem 3.4.1 If {v 1, v 2, …, v n } is a spanning set for a vector space V, then any collection of m vectors in V, where m>n, is linearly dependent. Corollary 3.4.2 If {v 1, v 2, …, v n } and {u 1, u 2, …, u m } are both bases for a vector space V, then n=m.

29 Definition Let V be a vector space. If V has a basis consisting of n vectors, we say that V has dimension n. The subspace {0} of V is said to have dimension 0. V is said to be finite- dimensional if there is a finite set of vectors that spans V; otherwise, we say that V is infinite-dimensional.

30 Theorem 3.4.3 If V is a vector space of dimension n>0: (1) Any set of n linearly independent vectors spans V. (2) Any n vectors that span V are linearly independent. Example Show that is a basis for R 3.

31 Theorem 3.4.4 If V is a vector space of dimension n>0, then: (1) No set of less than n vectors can span V. (2) Any subset of less than n linearly independent vectors can be extended to form a basis for V. (3) Any spanning set containing more than n vectors can be pared down to form a basis for V.

32 5 Change of Basis Changing Coordinates in R 2 x=x1e1+x2e2x=x1e1+x2e2 the coordinate of x is (x 1, x 2 ) T x=αy+βzx=αy+βzthe coordinate of x is (α, β) T

33 Changing Coordinates Let [e 1, e 2 ] be the standard basis, [u 1, u 2 ] is another basis. Two problems: (1) Given a vector x=(x 1, x 2 ) T, find its coordinates with respect to u 1 and u 2. (2) Given a vector c 1 u 1 +c 2 u 2, find its coordinates with respect to e 1 and e 2.

34 x=Uc the matrix U is called the transition matrix from the ordered basis [u 1, u 2 ] to the basis [e 1, e 2 ]. Example Let u 1 =(3,2) T, u 2 =(1,1) T, and x=(7,4) T. Find the coordinates of x with respect to u 1 and u 2.

35 Example Let b 1 =(1,-1) T, b 2 =(-2,3) T. Find the transition matrix from [e 1, e 2 ] to [b 1, b 2 ] and determine the coordinates of x=(1,2) T with respect to [b 1, b 2 ]. Example Find the transition matrix corresponding to the change of basis from [v 1, v 2 ] to [u 1, u 2 ], where and

36 Change of Basis for a General Vector Space Definition Let V be a vector space and let E=[v 1, v 2, …, v n ] be an ordered basis for V. If v is any element of V, then v can be written in the form v=c 1 v 1 +c 2 v 2 + ‥‥ +c n v n where c 1, …, c n are scalars. Thus we can associate with each vector v a unique vector c=(c 1, c 2, …, c n ) T in R n. The vector c defined in this way is called the coordinate vector of v with respect to the ordered basis E and is denoted [v] E. The c i ’s are called the coordinates of v relative to E.

37 Example Let E=[v 1, v 2, v 3 ]=[(1, 1, 1) T, (2, 3, 2) T, (1, 5, 4) T ] F=[u 1, u 2, u 3 ]=[(1, 1, 0) T, (1, 2, 0) T, (1, 2, 1) T ] Find the transition matrix from E to F. If x=3v 1 +2v 2 -v 3 and y=v 1 -3v 2 +2v 3 find the coordinates of x and y with respect to the ordered basis F.

38 6 Row Space and Column Space Definition If A is an m × n matrix, the subspace of R 1 × n spanned by the row vectors of A is called the row space of A. The subspace of R m spanned by the column vectors of A is called the column space of A. Theorem 3.6.1 Two row equivalent matrices have the same row space.

39 Definition The rank of a matrix A is the dimension of the row space of A. Example Let Theorem 3.6.2 (Consistency Theorem for Linear Systems) A linear system Ax=b is consistent if and only if b is in the column space of A.

40 Theorem 3.6.3 Let A be an m × n matrix. The linear system Ax=b is consistent for every b ∈ R m if and only if the column vectors of A span R m. The system Ax=b has at most one solution for every b ∈ R m if and only if the column vectors of A are linearly independent. Corollary 3.6.4 An n × n matrix A is nonsingular if and only if the column vectors of A form a basis for R n.

41 Definition The dimension of the nullspace of a matrix is called the nullity of the matrix.

42 Theorem 3.6.5 ( The Rank-Nullity Theorem) If A is an m × n matrix, then the rank of A plus the nullity of A equals n. Example Let Find a basis for the row space of A and a basis for N(A). Verify that dim N(A)=n-r.

43 Theorem 3.6.6 If A is an m × n matrix, the dimension of the row space of A equals the dimension of the column space of A. Example Let Find a basis for the column space of A.

44 Example Find the dimension of the subspace of R 4 spanned by


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