EE240A - MAE 270A - Fall 2002 Review of some elements of linear algebra Prof. Fernando Paganini September 27, 2002 1 Linear spaces and mappings In this section we will intro duce some of the basic ideas in linear algebra. Our treatment is primarily intended as a review for the reader's convenience, with some additional fo cus on the geometric asp ects of the sub ject. References are given at the end of the chapter for more details at b oth intro ductory and ad- vanced levels. 1.1 Vector spaces The structure intro duced now will p ervade our course, that of a vector space , also called a linear space. This is a set that has a natural addition op eration de ned on it, together with scalar multiplication. Because this is suchanim- p ortant concept, and arises in a numb er of di erentways,itisworth de ning it precisely b elow. Before pro ceeding we set some basic notation. The real numbers will be p denoted by R, and the complex numb ers by C ; wewilluse j := 1forthe imaginary unit. Also, given a complex number z = x + jy with x; y 2 R: z = x jy is the complex conjugate; p 2 2 jz j = x + y is the complex magnitude; x = Rez is the real part. + We use C to denote the op en right half-plane formed from the complex num- + b ers with p ositive real part; C is the corresp onding closed half-plane, and the left half-planes C and C are analogously de ned. Finally, j R denotes the imaginary axis. We now de ne a vector space. In the de nition, the eld F can be taken here to be the real numb ers R, or the complex numb ers C . The terminology real vector space, or complex vector space is used to sp ecify these alternatives. 1 De nition 1. Suppose V is a nonempty set and F is a eld, and that operations of vector addition and scalar multiplication are de ned in the fol lowing way. a For every pair u, v 2V a unique element u + v 2V is assignedcal led their sum; b for each 2 F and v 2V,there is a unique element v 2V cal led their product. Then V is a vector space if the fol lowing properties hold for al l u, v , w 2 V , and al l , 2 F : i There exists a zero element in V , denotedby0,suchthatv +0 = v ; ii thereexistsavector v in V , such that v +v =0; iii the association u +v + w =u + v +w is satis ed; iv the commutativity relationship u + v = v + u holds; v scalar distributivity u + v = u + v holds; vi vector distributivity + v = v + v is satis ed; vii the associative rule v = v for scalar multiplication holds; viii for the unit scalar 1 2 F the equality 1v = v holds. Formally,a vector space is an additive group together with a scalar multiplica- tion op eration de ned over a eld F , whichmust satisfy the usual rules v{viii of distributivity and asso ciativity. Notice that b oth V and F contain the zero element, whichwe will denote by \0" regardless of the instance. Given two vector spaces V and V , with the same asso ciated scalar eld, 1 2 weuseV V to denote the vector space formed by their Cartesian pro duct. 1 2 Thus every elementofV V is of the form 1 2 v ;v where v 2V and v 2V : 1 2 1 1 2 2 Having de ned a vector space wenow consider a numb er of examples. Examples: Both R and C can be considered as real vector spaces, although C is more commonly regarded as a complex vector space. The most common example of n arealvector space is R =R R; namely, n copies of R. We represent n elements of R in a column vector notation 3 2 x 1 7 6 . n . 2 R ; where each x 2 R: x = 5 4 k . x n 2 n Addition and scalar multiplication in R are de ned comp onentwise: 2 3 2 3 x + y x 1 1 1 6 7 6 7 x + y x 2 2 2 6 7 6 7 n x + y = ; x = ; for 2 R; x; y 2 R : 6 7 6 7 . 4 5 4 5 . x + y x n n n n Identical de nitions apply to the complex space C . As a further step, consider mn the space C of complex m n matrices of the form 2 3 a a 11 1n 6 7 . A = : 4 5 . a a m1 mn mn Using once again comp onentwise addition and scalar multiplication, C is a real or complex vector space. We now de ne two vector spaces of matrices which will be central in our mn course. First, we de ne the transp ose of the ab ove matrix A 2 C by 3 2 a a 11 m1 7 6 . nm 0 . 2 C ; A = 5 4 . a a 1n mn and its Hermitian conjugate or adjoint by 3 2 a a 11 m1 7 6 . nm . 2 C : A = 5 4 . a a mn 1n In b oth cases the indices have b een transp osed, but in the latter we also take the complex conjugate of each element. Clearly b oth op erations coincide if the matrix is real; wethus favor the notation A ,which will serve to indicate b oth 1 the transp ose of a real matrix, and the adjoint of a complex matrix. nn The square matrix A 2 C is Hermitian or self-adjoint if A = A : n The space of Hermitian matrices is denoted H , and is a real vector space. If a nn Hermitian matrix A is in R it is more sp eci cally referred to as symmetric. n The set of symmetric matrices is also a real vector space and will b e written S . m n n The set F R ; R of functions mapping m real variables to R is a vector space. Addition b etween two functions f and f is de ned by 1 2 f + f x ;::: ;x =f x ;::: ;x +f x ;::: ;x 1 2 1 m 1 1 m 2 1 m 1 0 The transp ose, without conjugation, of a complex matrix A will b e denoted by A ;how- ever, it is seldom required. 3 for anyvariables x ;::: ;x ; this is called p ointwise addition. Scalar multipli- 1 m cation byarealnumber is de ned by f x ;::: ;x = f x ;::: ;x : 1 m 1 m An example of a less standard vector space is given by the set comp osed of multinomials in m variables, that have homogeneous order n. We denote this [n] set by P . To illustrate the elements of this set consider m 2 3 p x ;x ;x =x x x ; p x ;x ;x =x x ; p x ;x ;x =x x x : 1 1 2 3 2 3 2 1 2 3 2 3 1 2 3 1 2 3 1 1 Each of these is a multinomial in three variables; however, p and p have order 1 2 [4] four, whereas the order of p is three. Thus only p and p are in P . Similarly 3 1 2 3 of 4 3 2 p x ;x ;x =x + x x and p x ;x ;x =x x x + x 4 1 2 3 2 5 1 2 3 2 3 1 1 3 1 [4] [n] only p is in P , whereas p is not in any P space since its terms are not 4 5 3 3 [n] homogeneous in order. Some thought will convince you that P is a vector m space under p ointwise addition. 1.2 Subspaces A subspace of a vector space V is a subset of V which is also a vector space with resp ect to the same eld and op erations; equivalently, it is a subset which is closed under the op erations on V . Examples: Avector space can havemany subspaces, and the simplest of these is the zero subspace, denoted by f0g. This is a subspace of anyvector space and contains only the zero element. Excepting the zero subspace and the entire space, the simplest typ e of subspace in V is of the form S = fs 2V : s = v ; for some 2 Rg; v given v in V . That is, each elementinV generates a subspace bymultiplying 2 3 it by all p ossible scalars. In R or R ,such subspaces corresp ond to lines going through the origin. Going back to our earlier examples of vector spaces we see that the multi- [n] m nomials P are subspaces of F R ; R, for any n. m n Now R has many subspaces and an imp ortant set is those asso ciated with m n the natural insertion of R into R ,whenm<n. Elements of these subspaces are of the form x ; x = 0 m nm wherex 2 R and 0 2 R . 4 Given two subspaces S and S we can de ne the addition 1 2 S + S = fs 2V : s = s + s for some s 2S and s 2S g 1 2 1 2 1 1 2 2 which is easily veri ed to b e a subspace. 1.3 Bases, spans, and linear indep endence Wenow de ne some key vector space concepts.