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Chapter 3 Vector Spaces

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  1. Chapter 3 Vector Spaces • 3.1 Vectors in Rn • 3.2 Vector Spaces • 3.3 Subspaces of Vector Spaces • 3.4 Spanning Sets and Linear Independence • 3.5 Basis and Dimension • 3.6 Rank of a Matrix and Systems of Linear Equations • 3.7 Coordinates and Change of Basis The idea of vectors dates back to the early 1800’s, but the generality of the concept waited until Peano’s work in 1888. It took many years to understand the importance and extent of the ideas involved. The underlying idea can be used to describe the forces and accelerations in Newtonian mechanics and the potential functions of electromagnetism and the states of systems in quantum mechanics and the least-square fitting of experimental data and much more.

  2. a point a vector 3.1 Vectors in Rn The idea of a vector is far more general than the picture of a line with an arrowhead attached to its end. A short answer is “A vector is an element of a vector space”. • Vector in Rn is denoted as an ordered n-tuple: which is shown to be a sequence of n real number • n-space: Rn is defined to be the set of all ordered n-tuple (1) An n-tuple can be viewed as a point in Rn with the xi’s as its coordinates. (2) An n-tuple can be viewed as a vector in Rnwith the xi’s as its components. • Ex:

  3. Note: A vector space is some set of things for which the operation of addition and the operation of multiplication by a scalar are defined. You don’t necessarily have to be able to multiply two vectors by each other or even to be able to define the length of a vector, though those are very useful operations. The common example of directed line segments (arrows) in 2D or 3D fits this idea, because you can add such arrows by the parallelogram law and you can multiply them by numbers, changing their length (and reversing direction for negative numbers).

  4. A complete definition of a vector space requires pinning down these properties of the operators and making the concept of vector space less vague. A vector space is a set whose elements are called “vectors” and such that there are two operations defined on them: you can add vectors to each other and you can multiply them by scalars (numbers). These operations must obey certain simple rules, the axioms for a vector space.

  5. (two vectors in Rn) • Equal: • if and only if • Vector addition (the sum of u and v): • Scalar multiplication (the scalar multiple of u by c):

  6. Zero vector: Notes: (1) The zero vector 0 in Rn is called the additive identityin Rn. (2) The vector –v is called the additive inverse of v. • Negative: • Difference:

  7. Thm 3.1: (the axioms for a vector space) Let v1, v2, and v3be vectors inRn , and let ,  and  be scalars.

  8. Ex : (Vector operations in R4) Let u=(2, – 1, 5, 0), v=(4, 3, 1, – 1), and w=(– 6, 2, 0, 3) be vectors in R4. Solve x for 3(x+w) = 2u –v+x Sol:

  9. Thm 3.2: (Properties of additive identity and additive inverse) Let v be a vector inRn and c be a scalar. Then the following is true. (1) The additive identity is unique. That is, if u+v=v, then u = 0 (2) The additive inverse of v is unique. That is, if v+u=0, then u = –v

  10. (1) 0v=0 (2) c0=0 (3) If cv=0, then c=0 or v=0 (4) (-1)v = -v and –(–v) = v • Thm 3.3: (Properties of scalar multiplication) Let vbe any element of a vector space V, and let c be any scalar. Then the following properties are true.

  11. Notes: A vector in can be viewed as: a 1×n row matrix (row vector): or a n×1 column matrix (column vector): (The matrix operations of addition and scalar multiplication give the same results as the corresponding vector operations)

  12. Vector addition Scalar multiplication Matrix Algebra

  13. is called a vector space Notes: (1) A vector space consists of four entities: a set of vectors, a set of scalars, and two operations V:nonempty set c:scalar vector addition scalar multiplication zero vector space containing only additive identity (2)

  14. Examples of vector spaces: (1) n-tuple space:Rn vector addition scalar multiplication (2) Matrix space: (the set of all m×n matrices with real values) Ex::(m = n = 2) vector addition scalar multiplication

  15. (3) n-th degree polynomial space: (the set of all real polynomials of degree n or less) (4) Function space:The set of square-integrable real-valued functions of a real variable on the domain [axb]. That is, those functions with . simply note the combination So the axiom-1 is satisfied. You can verify the rest 9 axioms are also satisfied.

  16. Function Spaces: Is this a vector space?How can a function be a vector? This comes down to your understanding of the word “function.” Is f(x) a function or is f(x) a number? Answer: It’s a number. This is a confusion caused by the conventional notation for functions. We routinely call f(x) a function, but it is really the result of feeding the particular value, x, to the function f in order to get the number f(x). Think of the function f as the whole graph relating input to output; the pair {x, f(x)} is then just one point on the graph. Adding two functions is adding their graphs.

  17. Ex1:The set of all integer is not a vector space. (it is not closed under scalar multiplication) Pf: scalar noninteger integer Ex2:The set of all second-degree polynomials is not a vector space. Pf: Let and (it is not closed under vector addition) Notes: To show that a set is not a vector space, you need only find one axiom that is not satisfied.

  18. Trivial subspace: Every vector space V has at least two subspaces. (1)Zero vector space {0} is a subspace of V. (2) V is a subspace of V. 3.3 Subspaces of Vector Spaces • Subspace: : a vector space : a nonempty subset :a vector space (under the operations of addition and scalar multiplication defined in V) Wis a subspace ofV

  19. Thm 3.4: (Test for a subspace) If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following conditions hold. (1) If u and v are in W, then u+v is in W. Axiom 1 (2) If uis in W and c is any scalar, then cuis in W. Axiom 2

  20. Ex: (A subspace of M2×2) Let W be the set of all 2×2 symmetric matrices. Show that W is a subspace of the vector space M2×2, with the standard operations of matrix addition and scalar multiplication. Sol:

  21. Ex: (Determining subspaces of R3) Sol:

  22. Thm 3.5: (The intersection of two subspaces is a subspace) Proof: Automatically from Thm 3.4.

  23. Given v = (– 1, – 2, – 2), u1= (0,1,4), u2= (– 1,1,2), and u3= (3,1,2) in R3, find a, b, and c such that v = 3.4 Spanning Sets and Linear Independence • Linear combination: Ex: Sol:

  24. Ex: (Finding a linear combination) Sol:

  25. (this system has infinitely many solutions)

  26. the span of a set: span (S) If S={v1, v2,…, vk} is a set of vectors in a vector space V, then the span of S is the set of all linear combinations of the vectors in S, • a spanning set of a vector space: If every vector in a given vector space U can be written as a linear combination of vectors in a given set S, then S is called a spanning set of the vector space U.

  27. Notes: • Notes:

  28. Ex: (A spanning set for R3) Sol:

  29. Thm 3.6: (Span (S) is a subspace of V) • If S={v1, v2,…, vk} is a set of vectors in a vector space V, • then • span (S) is a subspace of V. • span (S) is the smallest subspace of V that contains the spaning S. • i.e., • Every other subspace of V that contains S must contain span (S).

  30. Linear Independent (L.I.) and Linear Dependent (L.D.): : a set of vectors in a vector space V

  31. Notes:

  32. Ex: (Testing for linearly independent) Determine whether the following set of vectors in R3 is L.I. or L.D. Sol:

  33. c1(1+x – 2x2) + c2(2+5x –x2) + c3(x+x2) = 0+0x+0x2 i.e. c1+2c2 = 0 c1+5c2+c3 = 0 –2c1+ c2+c3 = 0    S is linearly dependent. (Ex: c1=2 , c2= – 1 , c3=3) • Ex: (Testing for linearly independent) Determine whether the following set of vectors in P2 is L.I. or L.D. S = {1+x – 2x2 , 2+5x –x2 , x+x2} v1v2v3 Sol: c1v1+c2v2+c3v3 = 0  This system has infinitely many solutions. (i.e., This system has nontrivial solutions.)

  34. Ex: (Testing for linearly independent) Determine whether the following set of vectors in 2×2 matrix space is L.I. or L.D. v1v2v3 Sol: c1v1+c2v2+c3v3 = 0

  35. 2c1+3c2+ c3 = 0 c1 = 0 2c2+2c3 = 0 c1+ c2 = 0   c1 = c2 = c3= 0 (This system has only the trivial solution.)  S is linearly independent.

  36. Thm 3.7: (A property of linearly dependent sets) A set S = {v1,v2,…,vk}, k2, is linearly dependent if and only if at least one of the vectors vj in S can be written as a linear combination of the other vectors in S. Pf: c1v1+c2v2+…+ckvk = 0  ci  0 for some i

  37. S is linearly dependent Let vi= d1v1+…+di-1vi-1+di+1vi+1+…+dkvk  d1v1+…+di-1vi-1+di+1vi+1-vi+…+dkvk = 0  c1=d1 , c2=d2 ,…, ci=-1 ,…, ck=dk (nontrivial solution) • Corollary to Theorem 3.7: Two vectors u and v in a vector space V are linearly dependent if and only if one is a scalar multiple of the other.

  38. 3.5 Basis and Dimension • Basis: Linearly Independent Sets V:a vector space Spanning Sets Bases S ={v1, v2, …, vn}V S spans V(i.e., span (S) = V ) S is linearly independent S is called a basis for V Bases and Dimension A basis for a vector space V is a linearly independent spanning set of the vector space V, i.e., any vector in the space can be written as a linear combination of elements of this set. The dimension of the space is the number of elements in this basis.

  39. Notes: (1) Ø is a basis for {0} (2) the standard basis for R3: {i, j, k} i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) • Note: • Beginning with the most elementary problems in physics and mathematics, it is clear that the choice of an appropriate coordinate system can provide great computational advantages. • For examples, • for the usual two and three dimensional vectors it is useful to express an arbitrary vector as a sum of unit vectors. • Similarly, the use of Fourier series for the analysis of functions is a very powerful tool in analysis. • These two ideas are essentially the same thing when you look at them as aspects of vector spaces.

  40. (4) the standard basis for mn matrix space: Ex: P3(x) { Eij | 1im , 1jn } {1, x, x2, x3} Ex: matrix space: (5) the standard basis for Pn(x): {1, x, x2, …, xn} (3) the standard basis for Rn : {e1, e2, …, en} e1=(1,0,…,0),e2=(0,1,…,0), en=(0,0,…,1) Ex: R4 {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}

  41. Span (S) = V Let v = c1v1+c2v2+…+cnvn • Thm 3.8: (Uniqueness of basis representation) If is a basis for a vector space V, then every vector in V can be written as a linear combination of vectors in S in one and only one way. Pf: • Span (S) = V • S is linearly independent v = b1v1+b2v2+…+bnvn  0 = (c1–b1)v1+(c2 –b2)v2+…+(cn–bn)vn  c1= b1 , c2= b2 ,…, cn= bn (i.e., uniqueness)

  42. Thm 3.9: (Bases and linear dependence) If is a basis for a vector space V, then every set containing more than n vectors in V is linearly dependent. Pf: Let S1 = {u1, u2, …, um} , m > n uiV

  43.  d1v1+d2v2+…+dnvn= 0 with di = ci1k1+ci2k2+…+cimkm Let k1u1+k2u2+…+kmum= 0  di=0 i i.e. If the homogeneous system (n<m) has fewer equations than variables, then it must have infinitely many solution. m > n k1u1+k2u2+…+kmum = 0 has nontrivial solution  S1 is linearly dependent

  44. dim(V) = n Linearly Independent Sets Spanning Sets Bases #(S) > n #(S) = n #(S) < n • Notes: (1) dim({0}) = 0 = #(Ø) (2) dim(V) = n , SV S:a spanning set  #(S)  n S:a L.I. set  #(S)  n S:a basis  #(S) = n (3) dim(V) = n , W is a subspace of V dim(W)  n

  45. Thm 3.10: (Number of vectors in a basis) If a vector space V has one basis with n vectors, then every basis for V has n vectors. i.e., All bases for a finite-dimensional vector space has the same number of vectors.) Pf: S ={v1, v2, …, vn} are two bases for a vector space S'={u1, u2, …, um}

  46. Dimension: The dimension of a finite dimensional vector space V is defined to be the number of vectors in a basis for V. V: a vector space S:a basis for V  dim(V) = #(S) (the number of vectors in S) • Finite dimensional: A vector space V is called finite dimensional, if it has a basis consisting of a finite number of elements. • Infinite dimensional: • If a vector space V is not finite dimensional, • then it is called infinite dimensional.

  47. (a) (d, c–d, c) = c(0, 1, 1) + d(1, – 1, 0) • Ex: (Finding the dimension of a subspace) (a) W1={(d, c–d, c): c and d are real numbers} (b) W2={(2b, b, 0): b is a real number} Sol: Find a set of L.I. vectors that spans the subspace.  S = {(0, 1, 1) , (1, – 1, 0)} (S is L.I. and S spans W1)  S is a basis for W  dim(W1) = #(S) = 2 (b)  S = {(2, 1, 0)} spans W2 and S is L.I.  S is a basis for W  dim(W2) = #(S) = 1

  48. spans W and S is L.I. • Ex: (Finding the dimension of a subspace) Let W be the subspace of all symmetric matrices in M22. What is the dimension of W? Sol:  S is a basis for W  dim(W) = #(S) = 3

  49. dim(V) = n Linearly Independent Sets Spanning Sets Bases #(S) > n #(S) < n #(S) = n • Thm 3.11: (Basis tests in an n-dimensional space) Let V be a vector space of dimension n. (1) If is a linearly independent set of vectors in V, then S is a basis for V. (2) If spans V, then S is a basis for V.