Linear Algebra, 5th Edition

1.5 Linear Dependence and Linear Independence

Suppose that V is a vector space over an infinite field and that W is a subspace of V. Unless W is the zero subspace, W is an infinite set. It is desirable to find a “small” finite subset S of W that generates W because we can then describe each vector in W as a linear combination of the finite number of vectors in S. Indeed, the smaller S is, the fewer the number of computations required to represent vectors in W as such linear combinations. Consider, for example, the subspace W of ${\text{R}}^3$ generated by $S=\{u_1,u_2,u_3,u_4\}$, where $u_1=(2,-1,4),u_2=(1,-1,3),u_3=(1,1,-1)$, and $u_4=(1,-2,-1)$. Let us attempt to find a proper subset of S that also generates W. The search for this subset is related to the question ...