Linear Vector Spaces - Usna.edu
Introduction to Linear Vector SpacesConcepts of primary interest:Closure and axioms for addition and multiplicationLinear independenceSpan, basis and dimensionInner ProductSample calculations:Span of a setCertifying a vector spaceCertifying an inner productGram-Schmidt basis constructionApplication examples:Legendre polynomialsTools of the trade:Desirable background:Fourier Series** Fourier series are used as examples. They should beskipped 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 mustidentify the definition of each concept. Do not blur the concept by using an exampleof the concept rather than its definition. You should be able to present prosestatements of definitions and axioms using complete sentences. You statementsshould be complete enough to remove the need for any other defining material suchas an equation. If you cannot do so, you do not know the definition. Memorizingdefinitions and statements of rules is the anchor step. Understanding cannot beachieved without it.Vector quantities play a central role in physics. At the lowest level, a vector is aquantity that has both magnitude and direction. A more complete and generalSend comments to: tank@alumni.rice.edu
description of vectors, vector spaces and their properties is to be developed in thissection and in a later section on linear transformations.The Displacement Model The properties of the collection of all possible displacements ri of a particle inour familiar model, a flat, infinite (Euclidean) three-dimensional universe, arepresented to introduce vector concepts.Rules for the Addition of DisplacementsA1. The sum of two displacements is another allowed displacement. r1 r2 r3A2. Additions can be regrouped without changing the resultant displacement. r1 r2 r3 r1 r2 r3A3. The order of additions can be changed without changing the sum. r1 r2 r2 r1A4. There is a displacement that does not change the position of the particle. ri 0 ri for all ri . A5. For any displacement ri , there is another ri that has the opposite action ona particle's position. ri ri 0 . for all ri ,Rules for the Multiplication of Displacements by Scalars M1. If ri is an allowed displacement, then any real scalar multiple of that displacement, c ri , is also an allowed displacement.M2. The scalar multiple of a sum of displacements is the same as the sum of thesame multiple of the individual displacements. c r1 r2 c r1 c r2 , an allowed displacementM3. A sum of scalars times a displacement is the sum of the individual scalarstimes that displacement. c d ri c ri d ri , an allowed displacement1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-2
M4. The scalar multiplication of displacements is associative. c d ri c d ri .M5. One times any displacement is that same displacement. 1 ri ri ri .The collection of all displacements has other features such as an inner (dot) productthat yields a scalar-valued measure of the component of one displacement along thedirection 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 isalways 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: ri ri ri ). Further,we can find a set of three displacements b , b , b such that all the allowed123displacements can be represented as a sum of multiples of the members of this set, a linear combination. That is: all ri c1 b1 c2 b2 c3 b3 for some b , b , b is a spanning set for our three dimensional universe. The minimum number of vectors b , b ,., b required such scalars: c1, c2 , c3 . This set of displacements1231 2n that every displacement can be represented as ri c1 b1 c2 b2 . cn b3 is thedimension of the space of displacements (three for our example). A spanning setwith the minimum required number of elements is a basis set. A common choice forthis set is {iˆ, ˆj , kˆ} .Graphical RepresentationsDropping back to two dimensions to simplify the drawings, the figure belowillustrates the concept of equality for displacements.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-3
yxAll 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 ratheris an adjustable parameter that may be used to be facilitate the solution of theproblem of interest. Move the displacement vector around without changing itsmagnitude or direction, and the action of the displacement is unchanged. (Moving avector without changing its direction is sometimes called parallel transport.) Vectoraddition illustrates the utility of the concept. r1 r2 r2 r1 r2 r2 r2 r1 r1The sum of two displacements should be the net displacement when the twodisplacements are executed in sequence. The left-hand figure illustrates transporting the tail of r2 to the tip of r1 to yieldthe sum of the displacements. The parallelogram rule diagram presents anothergraphical representation of vector addition.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-4
r1 r2 r1 r2 r1 r2 r2 r1Commutative Addition: Diagrams illustrating the commutative property (A3) for the addition of displacements in two dimensions. r1 r2 r2 r1Analytic (Component) MethodsGraphical techniques are perhaps superior for thinking about problems, but, for theheavy lifting, analytic methods are needed. A representation is adopted; acoordinate 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 ĵ 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 iˆ y ˆj . Once vector addition, scalar multiplication and the inner productoperations are defined in terms of these components, all problems can be submittedto a computer for analysis.y y r xiˆ y ˆj r ĵ y ˆj r y ˆj iˆ x 1/3/2009Physics Handout Series: Intro. Linear Vector Spaces xiˆ x VS-5
The component representation of a vector is found by representing the vector as a sum ofpieces with the number of pieces matching the dimension of the space and with each of thepieces running along one of the coordinate directions. In the figure above, one can dropperpendiculars to the x and y axes to fin the corresponding components. Notice that r x iˆ y ĵ is clearly a representation of r as the sum of a piece in the x directionplus a piece in the y direction.Component-based representation of: Addition: r1 r2 x1 iˆ y1 ˆj x2 iˆ y2 ˆj x1 x2 iˆ y1 y2 ĵ Scalar Multiplication: c r1 c x1iˆ y1 ˆj c x1 iˆ c y1 ˆj Inner Product: r1 r2 x1iˆ y1 ˆj x2iˆ y2 ˆj x1 x2 y1 y2 The operations of addition, scalar multiplication and inner product are defined interms of the scalar components of the vectors. The properties of the vectoroperations can now be based on the properties of operations on the scalarcomponents. The term 'scalars' means elements of a scalar field. Fields arediscussed briefly below.The direction cosines provide a particularly useful representation of the direction of avector. A direction has components that are the cosine of the angle between thatdirection and the coordinate direction of interest. eˆ iˆ eˆ iˆ ˆj eˆ ˆj kˆ eˆ kˆ cos e , x iˆ cos e, y ˆj cos e , z kˆEach direction cosine gauges the degree to which the direction ê points in a givencoordinate direction.The inner product is a gauge of common behavior. The innerproduct of two displacements gauges the degree to which thedisplacements are in the same direction and scales that result bythe product of the magnitudes of the individual displacements.This statement is in fuzzy-speak. It is not a math-worthy definition.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-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 anintroductory physics course. They apply to a number of physical vectors.Mathematicians collect the common rules from these behaviors and remove thecontext to define mathematical vector spaces. The spaces will include examples withvarious dimensions from one to infinity. The vector elements of these spaces can bequite different from our model of displacements in three dimensions. A space maybe the collection of all periodic functions with a period T or of all 7 x 7 matrices withcomplex elements.Don’t expect outcomes based on your prior experiences. Make your highestpriority memorizing the definitions of terms and statements of rules (axioms)as they arise. Base your arguments and conclusions on the new definitionsand rules rather than on any prejudices based on your understanding ofdisplacements.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 ofphysical or mathematical entities (elements) with defined binary addition and scalarmultiplication operations that satisfy the axioms below. The scalar multipliers arechosen from a field. The real numbers and complex numbers are examples of scalarfields. Just to ensure that you stay abstract and that you can't identify the vectorsdiscussed with any that you may have encountered in the past, vectors will berepresented as kets.combination A,B , C , . . The name ket is chosen because theis a bracket. It must follow that A is a bra and that Bis aket. A bra and ket A B together represents the inner product of the two vectors.A collection of entities (elements) forms a vector spaceif and only if the entitieshave defined operations for binary addition (for combining two vectors to yield1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-7
another vector) and for multiplication of a vector by a scalar to yield another vectorthat are closed (that yield results which are also in A1 & M1 below) and thatobey the rules: A2-A5 & M2-M5. The property blocks below first state the closureproperty and then list four more axioms satisfied by the addition and multiplicationprocesses respectively.TABLE I: Vector Space Axioms for Addition and MultiplicationAxioms for Vector Additionfor elements of a vector spaceA1. Closure under vector addition:For all A , B , A B C A2. The addition is associative.A B C A B CA3. The addition is commutative.A B B AA4. There is an additive identity (zero) vector.A 0 A for all A A5. There is an additive inverse for each vector.For all A A such that A A 0Axioms for Multiplication by a ScalarM1. Closure: The product of a scalar c and any vector inis also in.For all A V, c A VM2. Scalar multiplication is distributive across a sum of vectors.c A B c B c AM3. Scalar multiplication is distributive across a scalar sum. For scalars c and d, c d A c A d AM4. Scalar multiplication is associative. c k 1/3/2009A c k A Physics Handout Series: Intro. Linear Vector SpacesVS-8
M5. Action of the scalar identity1 A A for all A Learn to express each axiom in prose form!If only real scalarsscalarsare allowed, the space is a real linear vector space. If complexare allowed, then the vector space is a complex vector space. For ageneral case, the notation is to use a, b, c, . as elements of an unspecified scalarfield. Only spaces overandare of immediate interest.Closure example: Consider the addition of displacements. Start with S1, set of alldisplacements between points in an infinite three dimensional space. The sum oftwo such displacements is also in that set. Hence the set is closed under addition.Contrast this with S2, set of all displacements of magnitude 5 meters or less. Onecan add two displacements of magnitude 4 meters (which are members of that set)to get a displacement with a magnitude greater than 5 meters (which is not amember of the set). The second example set S2 is not closed under addition and sois not a candidate for certification as a vector space. Exercise: Consider the collection of objects of the form v a iˆ b ĵ where a and bare any real numbers. The vector addition and scalar (c) multiplication operationsusing real numbers are defined as: v1 v2 a1 a2 iˆ b1 b2 ˆj c v1 c a1iˆ b1 ˆj c a1 iˆ c b1 ˆj Show that the collection of all v as prescribed is a linear vector space. Assume thatscalar addition and multiplication of the real numbers obey all the common rules.Notation Alert: The use of the vector label notation 0 representing theadditive identity is sometimes preempted by another convention. For example,in quantum mechanics, 0 may represent the ground state (lowest energystate) for a problem. In these cases the additive identity is to be represented as1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-9
either ZERO or NULL . Stay alert; this conflict will arise when you study thequantum oscillatorWhat is a field? From: http://mathworld.wolfram.com/Field.htmlA field is any set of elements that satisfies the field axioms for both addition andmultiplication and is a commutative division algebra. An archaic name for a field isrational domain. The French term for a field is corps and the German word isKörper, both meaning "body." A field with a finite number of members is known as afinite field or Galois field.As the identity elements of a field must be different for addition and multiplication,every field (not vector space!) must have at least two elements. Examples of fieldsinclude the complex numbers, rational numbers, and real numbers, but not theintegers (as integers do not include multiplicative inverses for all integers).TABLE II: The field axioms written in addition-multiplication pairs.PropertyAdditionMultiplicationCommutativea b b aab baAssociativea (b c) (a b) ca (b c) (a b) cDistributivea (b c) a b a c( a b) c a c b cIdentitya 0 a1a aInversea ( a) 0a a 1 1(*)(*) The additive identity (0) is an exception; it does not have a multiplicative inverse.Notably, vector spaces lack self-multiplication and hence multiplicative inverses.Math Trivia: The vector space rules or axioms require that a space contain atleast one element, the additive null 0 . If it contains any other element A ,then it must include the additive inverse A and all elements of the form1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-10
c A where c is an element of the associated scalar field. All the fields that weconsider have an infinite number of elements so our vector spaces are likewiseinfinite, although they may be (and often are) finite dimensional.Linear dependence, spanning sets and basis sets:Defn: Linearly dependent set: A set of vectors ., Vi ,. is linearly dependent ifthere is a linear combination of those vectors that equals the zero vector,c1 V1 c2 V2 . ci Vi . 0 zeroand not all the scalars ci are zero.Suppose that c4 0. Then V4 can be represented as a sum of multiples of the othermembers of the set. That is V4 is linearly dependent on the other members of theset. If the zero vector 0 is a member of the set then the set is linearly dependentbecause the sum of one times 0 plus zero times the remaining members sums tothe zero vector 0 .If a set of vectors is not linearly dependent, it is linearly independent.The span of a set of vectors ., Vi ,. is all the vectors that can be representedas finite linear combinations of the members of the set ( ai Vi where the ai arearbitrary scalar coefficients).The set ., Vi ,. is said to span the space of elements ., ai Vi ,. for any ai 1/3/2009 aiVi . Span( ., Vi ,. ).Physics Handout Series: Intro. Linear Vector SpacesVS-11
The span of a set of vectors meets the requirements to be a vector space. The rulesfor forming linear combinations assure the axioms for binary addition and scalarmultiplications are met. ., Si ,. complete enough thatA spanning set for a space: Any set of vectorsevery vector in that spaceis of the formThat is: If the span of the set aiSi is a spanning set for that space.is the entire space, thenis a spanning set for.A spanning set must include at least one vector embodying each distinctcharacteristic behavior found in members of the space. A reduced spanning set isthe spanning set less one or more members which also spans the complete vectorspace. The requirement that the reduced set span the same space ensures that anymember that can be removed is a linear combination of the members that remain.Reducing a spanning set: Any members of the spanning set ., Vi ,. that arelinearly dependent on the other members of that spanning set can be removed from athe set without removing any vectors from the span of the set { .,the collection of all possible linear combinations of the members ofiVi , . },.A basis for a space: Any spanning set for a space that cannot be reduced is a basisset for that space. A basis is a spanning set with the smallest possible number ofelements. If a member is removed from a basis set, then the span of the remainingmembers no longer includes all the elements of the vector space. Equivalently, abasis set is a spanning set that is linearly independent.If a vector is removed from a basis set, at least one characteristic behaviorfound in the original space is lost.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-12
Dimension: The number of members in the smallest spanning set for a space is thedimension of that space. A set with the smallest number of members that stillspans the space is a basis for the space. This number (or dimension) is the sameas the largest number of linearly independent vectors in the space.The dimension of our space of all three-space displacements is, of course, three.A subspace is a vector spacevector spaceall of whose elements are contained in another. It follows that Dimension( ) Dimension( ). The span of any set isa vector space. If one vector is removed from a basis forremaining vectors in the set is a subspace ofthen the span of thewith dimension Dimension( ) -1.The concepts just presented are for general abstract vector spaces. A set ofconcrete examples is to be presented. Do not lose sight of the more generalnature of the concepts.Examples of the concepts restricted to 3D Cartesian displacement space.Linearly independent: The set of vectors {iˆ, ˆj kˆ, ˆj 2 kˆ} is linearly independent. It would be dependent if a iˆ b( ˆj kˆ) c( ˆj 2 kˆ) 0 for some {a,b,c }, not all zero. Thatwould require: a 0; b c 0, and b 2 c 0. Those conditions are contradictory.It can’t happen. The conditions require that a b c 0.Span of a set: The span of {iˆ, ˆj kˆ} is all vectors of the form a iˆ b( ˆj kˆ) or all thevectors in a plane that includes the x axis and the line y z. The span of{iˆ, ˆj kˆ, ˆj 2 kˆ} is all vectors of the form a iˆ b( ˆj kˆ) c( ˆj 2 kˆ) or all the vectors in theCartesian three-dimensional space.Spanning set: The set {iˆ, ˆj kˆ} is a spanning set for the space of all the vectors in aplane that includes the x axis and the line y z. The set {iˆ, ˆj kˆ, ˆj 2 kˆ} is a spanningset for the space of all the vectors in the Cartesian three-dimensional space. The set1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-13
{iˆ, ˆj kˆ, ˆj 2 kˆ, kˆ} is a spanning set for the space of all the vectors in the Cartesianthree-dimensional space. Note that the last spanning set contains more membersthan the dimension of the space that it spans.Basis set: A spanning set with the fewest number of members or equivalently that islinearly independent. The set {iˆ, ˆj kˆ} is a basis set for the space of all the vectorsin a plane that includes the x axis and the line y z. The set {iˆ, ˆj kˆ, ˆj 2 kˆ} is abasis set for the space of all the vectors in the Cartesian three-dimensional space.The set {iˆ, ˆj kˆ, ˆj 2 kˆ, kˆ} is a spanning set for the space of all the vectors in theCartesian three-dimensional space; it is not a basis set because the set is notlinearly independent (or has more members than the minimum required). Theminimum number of members is called the dimension of the space. Exercise: A vector is represented as v a iˆ b ( ˆj kˆ) c ( ˆj 2 kˆ) d kˆ , a linearcombination of a spanning set. Represent the vector as a combination of the members of a basis. v a iˆ b ( ˆj kˆ) c ( ˆj 2 kˆ) . What are a b and c The representation of a vector as a linear combination of members of a basis set isunique while there are many possible representations as linear combinations of themembers of a spanning set with more members than are found in a basis.Sample calculation of a span and identifying its dimension: Consider the set of vectors B1 2 iˆ 2 ˆj 2 kˆ, B2 4 iˆ 3 ˆj 3 kˆ, B3 ˆj kˆ over the real numbers . The span is the collection of all linear combinations of the three members of the set. Span(S) a 2 iˆ 2 ˆj 2 kˆ b 4 iˆ 3 ˆj 3 kˆ c ˆj kˆ a, b, c .Defining a 2 c and b c , Span(S) 2 4 iˆ 2 3 ˆj 2 3 kˆ , The last form is a curious rewriting, but it demonstrates that the Span( ) dependson only two free parameters and therefore must be suspected to be only two1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-14.
dimensional. Further B3 2 B1 B2 . That is B3 is linear combination of B1 and B2hence the set is linearly dependent. Span(S) a B1 b B2 c 2 B1 B2 B1 B2 a, b, c, , The space is two-dimensional as two vectors form a smallest spanning set for it. Exercise: Consider the set of vectors, iˆ ˆj kˆ, iˆ ˆj, kˆ .Display the general form forthe span of this set. Assume a space over the real numbers. The set is linearlydependent. Express k̂ as a linear combination of the other two vectors. Display areduced set that spans the same space as does the original set. Display a basis setfor the space spanned by iˆ ˆj kˆ, iˆ ˆj, kˆ .What is the dimension of that space?Describe the locus of points described if the vectors are considered to be a set ofdisplacements from the origin.The locus of points designated by the vectors spanned satisfies the equation y - x 0, aplane perpendicular to the x-y plane (as z is unrestricted) and intersecting with it on the liney x. The normal to the plane lies along the cross product of any two (non-parallel) vectors ˆj iˆ in the plane. Here iˆ ˆj kˆ iˆ ˆj ˆj iˆ , so one normal to the plane is nˆ . 2A Few Examples of Vector Spaces:N-tuple space. For a fieldwith members a, b, c, . consider the set of orderedcollections of scalar values A a1 , a2 ,., aN 1 , aN where addition is defined asA B a1 b1 , a2 b2 ,., aN 1 bN 1 , aN bN and multiplication is defined asc A c a1 , c a2 ,., c aN 1 , c aN . A basis for the space is {(1,0, .,0); (0,1,0, ,0), ,(0,0, ,1)}. No application has been discovered for this space, and its only habitatis mathematics texts. (Dimension N.)1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-15
3-D displacement space. For the field of the real numbers with members, a, b, c, d, . consider the ri ai iˆ bi ˆj ci kˆ where addition is defined as ri rj ai a j iˆ bi b j ˆj ci c j kˆ and multiplication is defined as d ri d ai iˆ d bi ˆj d ci kˆ . A basis for the space is { iˆ, ˆj , kˆ }. Examples of this spaceare common in physics texts.Polynomial (nth order) space. For the field of the real numbersor complexnumbers with members, a, b, c, d, . consider the PAn ( x) ao a1 x . an x n whereaddition is defined as PAn ( x) PBn ( x) ao bo a1 b1 x . an bn x n andmultiplication is defined as d PAn ( x) d ao d a1 x . d an x n . (Dimension n 1.) Abasis for the space is {x0, x1, , xn}.Taylor's series space. For the field of the real numbersor complexnumbers with members, a, b, c, d, . consider the TAn ( x) an x n where addition is defined asn 0 n 0n 0TAn ( x) TBn ( x) an bn x n and multiplication is defined as d TAn ( x) d an x n .The space is to be restricted to series that converge. (Dimension infinite.) A basisfor the space is {x0, x1, , xn, . , .}.Oscillator space. For the field of the real numberswith members, a, b, c, d, .consider the xi (t ) ai cos t bi sin t where addition is defined asxi (t ) x j (t ) ai a j cos t bi b j sin t and multiplication is defined asd xi (t ) d ai cos t d bi sin t .(Dimension homework.) A basis for thespace is {cos( t), sin( t)}.Fourier series space. For the field of the real numbersor complex numberswith members, a, b, c, d, . consider the Fourier space of elements1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-16
FS f c f 0 1 an 1fncos n 0t bm 1fmsin m 0t where addition is defined bysumming the corresponding Fourier coefficients and multiplication by d is definedby multiplying each Fourier coefficient by d.(Dimension infinite.) A basis forthe space is {1, { , cos(n t), },{ . , sin(m t), }} where m and n are strictlypositive integers.Legendre Polynomial Space for f( ) defined for 0 . For the field of the real numberswith members, a consider the Legendre space ofwell-behaved (finite-valued) elements for 0 : LS f f ( ) a 0f P (c os ) whereaddition is defined by summing the corresponding expansion coefficients andmultiplication by d is defined by multiplying each expansion coefficient by d.(Dimension infinite.) The basis is the set of P (cos ) for a positive integer. General Fourier series space. For the field of the complexnumbers with members a, b, c, d, . consider a Fourier space of elements i aim m ( x)wherem 1the ., m ( x),. are a set of linearly independent (preferably orthogonal) functions.For example, i ( x) ., m ,. might be the wave function for an electron in hydrogen. Thewould be the set of distinct stationary state wavefunctions for an electronin hydrogen. The state 1 could be the 1s ground state with energy -13.6 eV andspin up. Suggest definitions for addition and scalar multiplication for this space.The basis is the set ., m ( x),. of linearly independent (preferably orthogonal)functions.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-17
The Space of Null: A space containing only one element 0 , the zero or additiveidentity element. Note: space of null is the "dull space". The basis set for the spaceis the single element 0 . Certifying a vector space:A vector space is a set of elements that has as defined operations a binary additionand a scalar multiplication that satisfy the 10 axioms required of vector spaceoperation.Step 1: Identify the members of the set that is to be investigated. That identificationshould include what is essentially a rule to test for membership in the set.Step 2: The set must support two operations on it members, binary addition (thecombining of two members to yield a third) and scalar multiplication of eachmember to yield a member. The operations must be defined. The scalar fieldinvolved should be specified in this step.Step 3: Verify that each of the defined operations satisfies each of its five axioms. Isuggest beginning with closure for each operation. If the vectors have a componentrepresentation, the operations are usually defined in terms of operations on thescalar components which satisfy the field axioms. There is a field axiom that isparallel to each vector space axiom. Use it to establish that the vector space axiomis satisfied.The certification process should be exhaustive and exhausting. If it fails to be so,the certification is probably incomplete.SAMPLE CALCULATION: Certifying a set with its field and operations as a vector space:Show that the set of ordered 3-tuples of real numbers satisfies the requirements tobe a vector space with respect to the scalar field, the real numbers.1/3/2009Physics Handout Series: Intro. Linear Vector SpacesVS-18
The game plan is to note that the vector space operations are defined in terms ofparameters that are members of a scalar field. The operations of addition andmultiplication are defined in terms of the operations for that scalar field. Scalarfields satisfy axioms similar to those that are to be established in order that the setof objects be certified as a vector space. In fact, a scalar field is itself a onedimensional vector space with one or two extra features. For the case of a field, themultiplication is an operation on two members of the same set instead of one fromthe vector set and one from a separate scalar set, and a multiplicative inverse existsas a member of the set for all elements in the set except for the additive identity(zero).Set the stage by reviewing the contestan
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
Why Vector processors Basic Vector Architecture Vector Execution time Vector load - store units and Vector memory systems Vector length - VLR Vector stride Enhancing Vector performance Measuring Vector performance SSE Instruction set and Applications A case study - Intel Larrabee vector processor
The basic objects to be considered here are vector spaces of linear trans- formations, that is, a pair of vector spaces V and W and a linear subspace . The description of vector spaces of transformations of rank 1 is classical, . note will be concerned. Given any (abstract) vector space M, of dimension m, say, we may use the multiplication .
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Appendix A:Sample Parking Garage Operations Manual: Page A-4 1.4 Parking Facility Statistics Total Capacity: 2,725 Spaces Total Compact: 464 Spaces (17%) Total Full Size: 2,222 Spaces Total Handicapped Spaces: 39 Spaces Total Reserved Spaces: 559 Spaces Total Bank of America Spaces: 2,166 Spaces Total Boulevard Reserved
Vector Length (MVL) VEC-1 Typical MVL 64 (Cray) Add vector Typical MVL 64-128 Range 64-4996 (Vector-vector instruction shown) Vector processing exploits data parallelism by performing the same computation on linear arrays of numbers "vectors" using one instruction. The maximum number of elements in a vector supported by a vector ISA is
5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated .
Abstract: Here we define . all linear spaces of linear transformations over the field F. To define matrix addition in ( ), firstly we embed the given matrices into matrices of suitable higher order, and then . the set ( ) of all linear transformations from any vector space to any vector space, i.,e., the set ( ) forms a weak hemi-vector .
Tulang rawan yang paling banyak dijumpai pada orang dewasa. Lokasi : - Ujung ventral iga - Larynx,trachea, bronchus - Permukaan sendi tulang - Pada janin & anak yg sedang tumbuh pada lempeng epifisis Matriks tulang rawan hilain mengandung kolagen tipe II, meskipun terdapat juga sejumlah kecil kolagen tipe IX, X, XI dan tipe lainnya. Proteoglikan mengandung kondroitin 4-sulfat, kondroitin 6 .
