Advanced Calculus fi..

Advanced Calculus. Fifth EditionFigure 1.9Expression of u4 as a linear combination of u,, u2, u,.+ ~ 3 ~ 3be represented as clul + ~ 2 ~ 2 for appropriate el, ~ 2 c3, , as in the figure; thisis analogous to the representation of v in terms of i, j, k in Eq. (1.3) and Fig. 1.2.Nowso that again ul, . . . , ~q are linearly dependent.Accordingly, there cannot be four linearly independent vectors in space. Bysimilar reasoning we see that for every k greater than 3 there is no set of k linearlyindependent vectors in space.However, for k 5 3 there are k linearly independent vectors in space. Forexample, i, j is such a set of two vectors, and i, j, k is such a set of three vectors.(We can also consider i by itself-or any nonzero vector-as a set of one linearlyindependent vector.)Every triple ul , uz, u3 of linearly independent vectors in space serves as a basisfor vectors in space; that is, every vector in space can be expressed uniquely as alinear combination clul + c2u2 + c3u3, as in Fig. 1.9.We call i, j, k the standard basis. The equation v = v,i + v,j + v,k is therepresentation of v in terms of the standard basis.We observe that one could specialize the discussion of linear independenceto two-dimensional space-that is, the xy-plane. Here there are pairs of linearlyindependent vectors, and each such pair forms a basis; i, j is the standard basis.Every set of more than two vectors in the plane is linearly dependent.Planes in space. If PI: (xl, yl, zl) is a point of a plane and n = Ai + Bj + Ck is anonzero normal vector (perpendicular to the plane), then P: (x, y, z) is in the planeprecisely when(see Fig. 1.10). Equation (1.24) can be written as a linear equation