TAILIEUCHUNG - Finite Dimensional Vector Spaces - Halmos 2

(BQ) Finite dimensional vector spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. | CHAPTER HI ORTHOGONALITY 59. Inner products Let us now get our feet back on the ground. We started in Chapter I by pointing out that we wish to generalize certain elementary properties of certain elementary spaces such as R2. In our study so far we have done this but we have entirely omitted from consideration one aspect of R2. We have studied the qualitative concept of linearity what we have entirely ignored are the usual quantitative concepts of angle and length. In the present chapter we shall fill this gap we shall superimpose on the vector spaces to be studied certain numerical functions corresponding to the ordinary notions of angle and length and we shall study the new structure vector space plus given numerical function so obtained. For the added depth of geometric insight we gain in this way we must sacrifice some generality throughout the rest of this book we shall have to assume that the underlying field of scalars is either the field R of real numbers or the field e of complex numbers. For a clue as to how to proceed we first inspect R2. If X fl Ỉ2 and y i i ife are any two points in R2 the usual formula for the distance between X and y or the length of the segment joining X and y is V fl J 1 2 Ỉ2 I z 2. It is convenient to introduce the notation IIM Vfi2 fz2 for the distance from X to the origin 0 0 0 in this notation the distance between X and y becomes II X y II. So much for the present for lengths and distances what about angles It turns out that it is much more convenient to study in the general case not any of the usual measures of angles but rather their cosines. Roughly speaking the reason for this is that the angle in the usual picture in the cừcle of radius one is the length of a certain circular arc whereas the CO-118 Sec. 59 INNER PRODUCTS ĨĨ9 sine of the angle is the length of a line segment the latter is much easier to relate to our preceding study of linear functions. Suppose then that we let a be the angle between the segment from 0 to X

• Toán học
• Finite dimensional vector spaces
• Vector spaces
• Finite dimensional vector
• Modern mathematic
• Social sciences
• Population genetics
• Weather problems
• Electronic circuits
• Traffic flow
• Advanced calculus
• The differential calculus
• Scalar product spaces