By Hefferon J.

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**Example text**

Ck ∈ R} is generated by or spanned by the set of vectors {β1 , . . , βk }. There is a tricky point to this. We rely on the convention that the sum of an empty set of vectors is the zero vector. In particular, we need this in the case where a homogeneous system has a unique solution. Then the homogeneous case fits the pattern of the other solution sets: in the proof above, the solution set is derived by taking the c’s to be the free variables and if there is a unique solution then there are no free variables.

7 Intersect each pair, if possible. 8 When a plane does not pass through the origin, performing operations on vectors whose bodies lie in it is more complicated than when the plane passes through the origin. 5, 0, 1). 5, 1, 0). The endpoint of your vector is not (3, 2, 0); what is it? 5, 1, 0). The endpoint of the sum, on the diagonal, is not (3, 1, 1); what is it? 9 Show that the line segments (a1 , a2 )(b1 , b2 ) and (c1 , c2 )(d1 , d2 ) have the same lengths and slopes if b1 − a1 = d1 − c1 and b2 − a2 = d2 − c2 .

1 Vectors in Space “Higher-dimensional geometry” sounds exotic. It is exotic — interesting and eye-opening. But it isn’t distant or unreachable. We begin by defining one-dimensional space to be the set R1 . To see that definition is reasonable, draw a one-dimensional space and make the usual correspondence with R: pick a point to label 0 and another to label 1. 17 times as far. Section II. Linear Geometry of n-Space 33 The basic idea here, combining magnitude with direction, is the key to extending to higher dimensions.