Linear Algebra with Applications, 10th Edition

Chapter 5 Orthogonality

Full Alternative Text

We can add to the structure of a vector space by defining a scalar or inner product. Such a product is not a true vector multiplication, since to every pair of vectors it associates a scalar rather than a third vector. For example, in ${ℝ}^2$, we can define the scalar product of two vectors x and y to be ${\mathbf{x}}^T\mathbf{y}$. We can think of vectors in ${ℝ}^2$ as directed line segments beginning at the origin. It is not difficult to show that the angle between two line segments will be a right angle if and only if the scalar product of the corresponding vectors is zero. In general, if V is a vector space with a scalar ...