Page 448 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution (b) Figure 6.1.1
, so the equation of the unit circle is
If , then , or, on squaring both
sides,

The graph of this equation is the ellipse shown in Figure 6.1.1b.

It would be reasonable for you to feel uncomfortable with the results in the last example, because although our definitions of
length and distance reduce to the standard definitions when applied to with the Euclidean inner product, it does require a stretch
of the imagination to think of the unit “circle” as having an elliptical shape. However, even though nonstandard inner products
distort familiar spaces and lead to strange values for lengths and distances, many of the basic theorems of Euclidean geometry
continue to apply in these unusual spaces. For example, it is a basic fact in Euclidean geometry that the sum of the lengths of two
sides of a triangle is at least as large as the length of the third side (Figure 6.1.2a). We shall see later that this familiar result holds
in all inner product spaces, regardless of how unusual the inner product might be. As another example, recall the theorem from
Euclidean geometry that states that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of
the four sides (Figure 6.1.2b). This result also holds in all inner product spaces, regardless of the inner product (Exercise 20).

Figure 6.1.2

Inner Products Generated by Matrices