The elements x + y and ax are called the sum of x and y and the of a and x, respectively. (VS 8) For each pair of elements o, b in F and each clement x in V, (a + b)x - ax + bx. (VS 7) For each clement a in F and each pair of elements x, y in V, a(x + y) - ax + ay. (VS 6) For each pair of elements a, b in F and each element x in V, (ab)x - a(bx). (VS 4) For each element x in V there exists an clement y in V such that x + y = 0. (VS 3) There exists an element in V denoted by 0 such that x + 0 = x for each x in V. (VS 2) For all x, y, z in V, (x + y) + z = x + (y + z) (associativity of addition). (VS 1) For all x, y in V, x + y = y + x (commutativity In V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold. A vector space (or linear space) V over a field2 F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so that for each pair of elements x, y. In this section, we introduce some of these systems, but first we formally define this type of algebraic structure. Many other familiar algebraic systems also permit definitions of addition and scalar multiplication that satisfy the same eight properties. we saw that, with the natural definitions of vector addition and scalar" multiplication, the vectors in a plane satisfy the eight properties listed on page 3. Prove that the diagonals of a parallelogram bisect each other. Show that the midpoint of the line segment joining the points (a.b) and (c.d) is ((a \ c)/2,(6 + d)/2). page 82 page 9 page 295 age 295 age 67 >age 69 Piage 8 ige 448 P 0 and in the opposite direction if t ) Page page page page page page page page page The limit of a sequence of matrices the space of linear transformations from V to V the space of linear transformations from V to W the set of m x // matrices with entries in F the column sum of the matrix A the j t h column sum of the matrix A the null space of T the dimension of the null space of T the zero matrix the permanent of the 2 x 2 matrix M the space of polynomials with coefficients in F the polynomials in P(F) of degree at most // standard representation with respect to basis 3 the field of real numbers the rank of the matrix A the rank of the linear transformation T the row sum of the matrix A the ith row sum of the matrix A the range of the linear transformation T CONTINUED ON REAR ENDPAPERS Left-multiplicatioD transformation by matrix A The iih Gerschgorin disk the condition number of the matrix A set of functions / on R with / ' " ' continuous set of functions with derivatives of every order The T-cyclic basis generated by x the field of complex numbers The ij-th entry of the matrix A the inverse of the matrix AĪdjoint of the matrix A matrix A with row i and column j deleted transpose of the matrix A matrix A augmented by the matrix B direct sum of matrices Bi through B^ set of bilinear forms on V