Linear Algebra, 5th Edition

1.4 Linear Combinations and Systems of Linear Equations

In Section 1.1, it was shown that the equation of the plane through three noncollinear points A, B, and C in space is $x=A+su+tv$, where u and v denote the vectors beginning at A and ending at B and C, respectively, and S and t denote arbitrary real numbers. An important special case occurs when A is the origin. In this case, the equation of the plane simplifies to $x=su+tv$, and the set of all points in this plane is a subspace of ${\text{R}}^3$. (This is proved as Theorem 1.5.) Expressions of the form $su+tv$, where s and t are scalars and u and v are vectors, play a central role in the theory of vector spaces. The appropriate generalization of such expressions is presented in the following definitions. ...