2.1 Vector-Vector Products Given two vectors r,ye R",the quantity ry,sometimes called the inner product or dot product of the vectors,is a real number given by 1 2 x'y∈R=[x12·cn : Tiyi i=l Observe that inner products are really just special case of matrix multiplication.Note that it is always the case that aTy=yTc. Given vectors E R,yE R(not necessarily of the same size),xyTE Rmxn is called the outer product of the vectors.It is a matrix whose entries are given by (y)=ij, i.e., T1 T141 T142 ··1yn xyT∈Rmxn T2 E21 E22 T2Un 1 2· m]= Imy1 Tmy2 TmYn As an example of how the outer product can be useful,let 1 ER"denote an n-dimensional vector whose entries are all equal to 1.Furthermore,consider the matrix A E Rmxn whose columns are all equal to some vector xE Rm.Using outer products,we can represent A compactly as, T1 --小 2 2 T2 2 [11 1]=x1 工m m 2.2 Matrix-Vector Products Given a matrix A E Rmxn and a vector x E R",their product is a vector y=Ax E Rm. There are a couple ways of looking at matrix-vector multiplication,and we will look at each of them in turn. If we write A by rows,then we can express Ax as, x Ax= 42.1 Vector-Vector Products Given two vectors x, y ∈ R n , the quantity x T y, sometimes called the inner product or dot product of the vectors, is a real number given by x T y ∈ R = x1 x2 · · · xn      y1 x2 . . . yn      = Xn i=1 xiyi . Observe that inner products are really just special case of matrix multiplication. Note that it is always the case that x T y = y T x. Given vectors x ∈ R m, y ∈ R n (not necessarily of the same size), xyT ∈ R m×n is called the outer product of the vectors. It is a matrix whose entries are given by (xyT )ij = xiyj , i.e., xyT ∈ R m×n =      x1 x2 . . . xm      y1 y2 · · · yn =      x1y1 x1y2 · · · x1yn x2y1 x2y2 · · · x2yn . . . . . . . . . . . . xmy1 xmy2 · · · xmyn      . As an example of how the outer product can be useful, let 1 ∈ R n denote an n-dimensional vector whose entries are all equal to 1. Furthermore, consider the matrix A ∈ R m×n whose columns are all equal to some vector x ∈ R m. Using outer products, we can represent A compactly as, A =   | | | x x · · · x | | |   =      x1 x1 · · · x1 x2 x2 · · · x2 . . . . . . . . . . . . xm xm · · · xm      =      x1 x2 . . . xm      1 1 · · · 1 = x1 T . 2.2 Matrix-Vector Products Given a matrix A ∈ R m×n and a vector x ∈ R n , their product is a vector y = Ax ∈ R m. There are a couple ways of looking at matrix-vector multiplication, and we will look at each of them in turn. If we write A by rows, then we can express Ax as, y = Ax =      — a T 1 — — a T 2 — . . . — a T m —      x =      a T 1 x a T 2 x . . . a T mx      . 4