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Z,a, = b, has at most one solution when the associated homogeneous system (HS) z1a1 + . . + z,a, = 0, only has the trivial solution 2 1 = 0, . . , z, = 0. This happens when no aj can be written as a linear combination of the other ai's. Indeed, if aj is a linear combination of the other ai's, say CHAPTER 2. VECTOR SPACES 40 then ( H S ) has a solution set with x j = 1, hence a nontrivial solution. It is natural to say that the m-tuples al, . . ,a, are linearly independent when none is a linear combination of the others.
P,-l, P, = Po such that Mi is the midpoint between Pi-1 and Pi (1 6 i n)? When it is possible, are there many possibilities? 9. Let < 10. Let P I , Pz,. . Is it always possible to find disjoint balls Bi with center Pi (1 i 6 n) such that Bi is tangent to both Bi-1 and where BO= B, and B,+1 = B1. The problem is to find the radii of these balls, as a function of the distance of consecutive Pi's. < 11. The equation of a plane in the usual space has the form ux + by + cz = d, where a , b, c, and d are parameters depending on the plane.
Ax,a, = axial+ a ~ a 2 (by the axioms of vector spaces, valid in E ) , and similarly, the sum of two linear combinations (xlal + . . + x,a,) + (ylal + . . + ynan) = (XI + y1)al + . . + (x, + yn)an, is again a linear combination. This subspace is called the linear span of the finite subset al, a2,. . ,a, (or of the family ( a i ) l G i G n ) , and denoted by V = L(al,a2,. . , a , ) = L(ai : 1 < a < n). It is the smallest subspace containing al, a 2 , . . ,a,, since any subspace W of El containing these elements, will also contain their linear combinations, hence contain L ( a l , a 2 , ..