We saw in Section 1.3 that there were various ways in which the geometry of could shed light on linear systems of equations. We used a very limited amount of geometry, though; we only made use of general vector space operations. The geometry of is much richer than that of an arbitrary vector space because of the concepts of length and angles; it is extensions of these ideas that we will explore in this chapter.
The Dot Product in
Recall the following definition from Euclidean geometry.
Definition Let. The dot product or inner product of x and y is denoted and is defined by
where x =and y =.
Quick Exercise #1. Show that for.
The dot product is intimately related to the ideas of length and angle: the length of a vector is given by
and the angle between two vectors x and y is given by
In particular, the dot product gives us a condition for perpendicularity: two vectors are perpendicular if they meet at a right angle, which by the formula above is equivalent to the condition. For example, the standard basis vectors are perpendicular to each other, since for.
Perpendicularity is an extremely useful concept in the context of linear algebra; the following proposition gives a first hint as to why.
Proposition 4.1 Let denote the standard basis of. If then for each.
Proof For v as above,
since has a 1 in the jth position and zeroes everywhere else
We will soon see that, while the computations are particularly easy with the standard basis, the crucial property that makes Proposition 4.1 work is the perpendicularity of the basis elements.
Inner Product Spaces
Motivated by the considerations above, we introduce the following extra kind of structure for vector spaces. Here and for the rest of the chapter, we will only allow the base field to be or. Recall that for, the complex conjugate of is defined by, and the absolute value or modulus is defined by.