1/45Review of Matrices and Vectors2/45Definition of Vector: A collection of complex or real numbers, generally put in a column[]TNNvvvv!"11==vTranspose++=+==NNNNbababbaa"""1111babaDefinition of Vector Addition: Add element-by-elementVectors & Vector Spaces3/45Definition of Scalar: A real or complex number. If the vectors of interest are complex valued then the set of scalars is taken to be complex numbers; if the vectors of interest are real valued then the set of scalars is taken to be real numbers.Multiplying a Vector by a Scalar :αα=α=NNaaaa""11aa…changes the vector’s length if |α| ≠ 1 … “reverses” its direction if α< 04/45Arithmetic Properties of Vectors: vector addition and scalar multiplication exhibit the following properties pretty much like the real numbers doLet x, y, and z be vectors of the same dimension and let αand β be scalars; then the following properties hold:ααxxxyyx=+=+xxzxyzyx)()()()(αβ=βα++=++xxxyxyxβ+α=β+αα+α=+α)()(zeros all of vector zero theis where,1000xxx==1. Commutativity 2. Associativity3. Distributivity4. Scalar Unity & Scalar Zero5/45Definition of a Vector Space: A set V of N-dimensional vectors (with a corresponding set of scalars) such that the set of vectors is:(i) “closed” under vector addition(ii) “closed” under scalar multiplicationIn other words:• addition of vectors – gives another vector in the set• multiplying a vector by a scalar – gives another vector in the setNote: this means that ANY “linear combination” of vectors in the space results in a vector in the space…If v1, v2, and v3are all vectors in a given vector space V, then∑==++=31332211iiivvvvvααααis also in the vector space V.6/45Axioms of Vector Space: If V is a set of vectors satisfying the above definition of a vector space then it satisfies the following axioms:1. Commutativity (see above)2. Associativity (see above)3. Distributivity (see above)4. Unity and Zero Scalar (see above)5. Existence of an Additive Identity – any vector space V must have a zero vector6. Existence of Negative Vector: For every vector v in V its negative must also be in VSo… a vector space is nothing more than a set of vectors with an “arithmeticstructure”7/45Def. of Subspace: Given a vector space V, a subset of vectors in Vthat itself is closed under vector addition and scalar multiplication (using the same set of scalars) is called a subspace of V.Examples: 1. The space R2is a subspace of R3. 2. Any plane in R3 that passes through the origin is a subspace3. Any line passing through the origin in R2is a subspace of R2 4. The set R2is NOT a subspace of C2because R2isn’t closed under complex scalars (a subspace must retain the original space’s set of scalars)8/45Length of a Vector (Vector Norm): For any vector v in CNwe define its length (or “norm”) to be ∑==Niiv122v∑==Niiv1222v22vvαα=2221221vvvvβαβα+≤+NC∈∀∞< vv20vv == iff02Properties of Vector Norm:Geometric Structure of Vector Space9/45Distance Between Vectors: the distance between two vectors in a vector space with the two norm is defined by: 22121),( vvvv −=d2121 iff 0),( vvvv ==dNote that:v2v1v1– v210/45Angle Between Vectors & Inner Product:Motivate the idea in R2:vθAu=θθ=01sincosuvAAθ=θ⋅+θ⋅=∑=cossin0cos121AAAvuiiiNote that:Clearly we see that… This gives a measure of the angle between the vectors. Now we generalize this idea!11/45Inner Product Between Vectors : Define the inner product between two complex vectors in CN by: *1iNiivu∑=>=< vu,Properties of Inner Products:1. Impact of Scalar Multiplication:2. Impact of Vector Addition:3. Linking Inner Product to Norm:4. Schwarz Inequality:5. Inner Product and Angle: (Look back on previous page!)><>=<><>=<vu,vu,vu,vu,*ββαα><+>>=<+<><+>>=<+<vw,vu,vw,uzu,vu,zvu,>=< vv,v2222vuvu, ≤><)cos(22θ=><vuvu,12/45Inner Product, Angle, and Orthogonality : )cos(22θ=><vuvu,(i) This lies between –1 and 1; (ii) It measures directional alikeness of u and v= +1 when u and v point in the same direction= 0 when u and v are a “right angle”= –1 when u and v point in opposite directionsTwo vectors u and v are said to be orthogonal when <u,v> = 0If in addition, they eachhave unit length they are orthonormal13/45Can we find a set of “prototype” vectors {v1, v2, …, vM} from which we can build all other vectors in some given vector space V by using linear combinations of the vi? Same “Ingredients”… just different amounts of them!!!∑==Mkkk1vvα∑==Mkkk1vuβWe want to be able to do is get any vector just by changing the amounts… To do this requires that the set of “prototype”vectors {v1, v2, …, vM} satisfy certain conditions.We’d also like to have the smallest number of members in the set of “prototype” vectors. Building Vectors From Other Vectors14/45Span of a Set of Vectors: A set of vectors {v1, v2, …, vM} is said to span the vector space V if it is possible to write each vector v in V as a linear combination of vectors from the set:∑=α=Mkkk1vvThis property establishes if there are enough vectors in the proposed prototype set to build all possible vectors in V.It is clear that:1. We need at least N vectors to span CNor RNbut not just any Nvectors.2. Any set of N mutually orthogonal vectors spans CNor RN(a set of vectors is mutually orthogonal if all pairs are orthogonal).Does not Span R2Spans R2Examples in R215/45Linear Independence: A set of vectors {v1, v2, …, vM} is said to be linearly independent if none of the vectors in it can be written as a linear combination of the others. If a set of vectors is linearly dependent then there is “redundancy” in the set…it has more vectors than needed to be a “prototype” set!For example, say that we have a set of four vectors {v1, v2, v3, v4} and lets say that we know that we can build v2from v1and v3…then every vector we can build from {v1, v2, v3, v4} can also be built from only {v1, v3, v4}.It is clear that:1. In CNor RNwe can have no more than N linear independent vectors.2. Any set of mutually orthogonal vectors is linear independent (a set of vectors is mutually orthogonal if all pairs are orthogonal).Linearly IndependentNot Linearly IndependentExamples