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Linear Vector Spaces Introduction to Linear Vector Spaces Concepts of primary interest: Closure and axioms for addition and multiplication Linear independence Span, basis and dimension Inner Product Sample calculations: Span of a set Certifying a vector space Certifying an inner product Gram-Schmidt basis construction Application examples: Legendre polynomials Tools of the trade: Desirable background: Fourier Series* * Fourier series are used as examples. They should be skipped if you are not familiar with Fourier series. Study Guide: Definitions and statements of fundamental rules (such as axioms) must be memorized as soon as one begins the study of a new topic. One must identify the definition of each concept. Do not blur the concept by using an example of the concept rather than its definition. You should be able to present prose statements of definitions and axioms using complete sentences. You statements should be complete enough to remove the need for any other defining material such as an equation. If you cannot do so, you do not know the definition. Memorizing definitions and statements of rules is the anchor step. Understanding cannot be achieved without it. Vector quantities play a central role in physics. At the lowest level, a vector is a quantity that has both magnitude and direction. A more complete and general Send comments to: [email protected] description of vectors, vector spaces and their properties is to be developed in this section and in a later section on linear transformations. The Displacement Model The properties of the collection of all possible displacements ri of a particle in our familiar model, a flat, infinite (Euclidean) three-dimensional universe, are presented to introduce vector concepts. Rules for the Addition of Displacements A1. The sum of two displacements is another allowed displacement. rr12 r3 A2. Additions can be regrouped without changing the resultant displacement. rrr123 rr 12 r3 A3. The order of additions can be changed without changing the sum. rr12 r 2 r1 A4. There is a displacement that does not change the position of the particle. ri0 ri for all ri . A5. For any displacement ri , there is another ri that has the opposite action on a particle's position. for all ri , ri ri 0 . Rules for the Multiplication of Displacements by Scalars M1. If ri is an allowed displacement, then any real scalar multiple of that displacement, cr i , is also an allowed displacement. M2. The scalar multiple of a sum of displacements is the same as the sum of the same multiple of the individual displacements. cr12 r crcr 1 2, an allowed displacement M3. A sum of scalars times a displacement is the sum of the individual scalars times that displacement. cd ri crdrii, an allowed displacement 1/3/2009 Physics Handout Series: Intro. Linear Vector Spaces VS-2 M4. The scalar multiplication of displacements is associative. cd ri cd ri . M5. One times any displacement is that same displacement. 1rri ii r. The collection of all displacements has other features such as an inner (dot) product that yields a scalar-valued measure of the component of one displacement along the direction of a second displacement times the magnitude of the second displacement. It gauges the degree to which the two displacements embody common behavior, displacing in the same direction. The inner product of a displacement with itself is always positive. Thus the inner product can be used to assign lengths to all the displacements in the space (to set a norm for the space: rri iri|). Further, we can find a set of three displacements bbb123,, such that all the allowed displacements can be represented as a sum of multiples of the members of this set, a linear combination. That is: all rci 11bcb 2 2 cb3 3 for some scalars: ccc123,, . This set of displacements bb12,,b3is a spanning set for our three dimensional universe. The minimum number of vectors bb12,,..., bn required such that every displacement can be represented as rcbi 11cb2 2...cbn 3 is the dimension of the space of displacements (three for our example). A spanning set with the minimum required number of elements is a basis set. A common choice for this set is{,ijkˆˆ ,ˆ }. Graphical Representations Dropping back to two dimensions to simplify the drawings, the figure below illustrates the concept of equality for displacements. 1/3/2009 Physics Handout Series: Intro. Linear Vector Spaces VS-3 y x All the displacements illustrated have the same magnitude (length) and direction; they are all equal. The 'location' of the displacement is not considered, and, in fact, the location of the vector is not an intrinsic property of the displacement, but rather is an adjustable parameter that may be used to be facilitate the solution of the problem of interest. Move the displacement vector around without changing its magnitude or direction, and the action of the displacement is unchanged. (Moving a vector without changing its direction is sometimes called parallel transport.) Vector addition illustrates the utility of the concept. r1+r2 r2 r1+r2 r2 r2 r1 r1 The sum of two displacements should be the net displacement when the two displacements are executed in sequence. The left-hand figure illustrates transporting the tail of r2 to the tip of r1 to yield the sum of the displacements. The parallelogram rule diagram presents another graphical representation of vector addition. 1/3/2009 Physics Handout Series: Intro. Linear Vector Spaces VS-4 r1 r1+r2 r2 r1+r2 r2 r1 Commutative Addition: Diagrams illustrating the commutative property (A3) for the addition of displacements in two dimensions. rr12 r 2 r1 Analytic (Component) Methods Graphical techniques are perhaps superior for thinking about problems, but, for the heavy lifting, analytic methods are needed. A representation is adopted; a coordinate system is chosen. The axes are chosen to be perpendicular (orthogonal). The values x and y are the projections of r onto the x and y axes. The unit (normalized basis) vectors iˆ and ˆj are chosen to have the directions of the x and y axes. The displacement r can then be represented in terms of its components as x iyˆˆj. Once vector addition, scalar multiplication and the inner product operations are defined in terms of these components, all problems can be submitted to a computer for analysis. y y rx ˆˆiyj r ˆj yˆj r yˆj iˆ x xiˆ x 1/3/2009 Physics Handout Series: Intro. Linear Vector Spaces VS-5 The component representation of a vector is found by representing the vector as a sum of pieces with the number of pieces matching the dimension of the space and with each of the pieces running along one of the coordinate directions. In the figure above, one can drop perpendiculars to the x and y axes to fin the corresponding components. Notice that rxiyˆˆj is clearly a representation of r as the sum of a piece in the x direction plus a piece in the y direction. Component-based representation of: ˆˆ ˆˆ ˆ ˆ Addition: rr12 xiyjxiyjxxiyy11 2 2 12 12j ˆˆ ˆ ˆ Scalar Multiplication: cr111 c xi yj cxi 1 cyj 1 ˆˆ ˆˆ Inner Product: rr12 xiyjxiyj 1 1 2 2 xx 12 yy 12 The operations of addition, scalar multiplication and inner product are defined in terms of the scalar components of the vectors. The properties of the vector operations can now be based on the properties of operations on the scalar components. The term 'scalars' means elements of a scalar field. Fields are discussed briefly below. The direction cosines provide a particularly useful representation of the direction of a vector. A direction has components that are the cosine of the angle between that direction and the coordinate direction of interest. ˆˆˆˆˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ eieijejkek cosex,, i cosey j cos ez, k Each direction cosine gauges the degree to which the direction eˆ points in a given coordinate direction. The inner product is a gauge of common behavior. The inner product of two displacements gauges the degree to which the displacements are in the same direction and scales that result by the product of the magnitudes of the individual displacements. This statement is in fuzzy-speak. It is not a math-worthy definition. 1/3/2009 Physics Handout Series: Intro. Linear Vector Spaces VS-6 The Big Shift: A summary of the properties of displacements has been presented. Those properties are those that are characteristic of vectors as studied in an introductory physics course. They apply to a number of physical vectors. Mathematicians collect the common rules from these behaviors and remove the context to define mathematical vector spaces. The spaces will include examples with various dimensions from one to infinity. The vector elements of these spaces can be quite different from our model of displacements in three dimensions. A space may be the collection of all periodic functions with a period T or of all 7 x 7 matrices with complex elements. Don’t expect outcomes based on your prior experiences. Make your highest priority memorizing the definitions of terms and statements of rules (axioms) as they arise. Base your arguments and conclusions on the new definitions and rules rather than on any prejudices based on your understanding of displacements. We will return to the concept of a physical vector in another section. At that time, the properties required to qualify an entity as a physical vector will be discussed. An Abstract Linear Vector Space: By definition, a vector space is any collection of physical or mathematical entities (elements) with defined binary addition and scalar multiplication operations that satisfy the axioms below.
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