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MAT237Y1 Study Guide - Scalar Multiplication, Parallelogram, Euclidean Space

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section 1.1 from Rto RnMAT 237 Notes
This course attempts at generalizing functions of one variable like y=f(x)to functions of several
variables, that is functions that take more than one variables as their argument (or their input:) denoted
by f(x, y), f(x, y, z)or f(x1, x2,...xn). So the expression f(x, y, z)stands for the value that fassigns
to the input (x, y, z). In multivariate Calculus we are trying to extend the main ideas and techniques of
one variable Calculus to the multi-variate functions. Working with functions of one variable we use many
famous properties of R. This may sometimes happen unconsciously. It is a common practice in one variable
Calculus to work with expressions such as f(x), f(x+h), or f(kx). These operations make sense since
the range of the variable xis real numbers, and in the real numbers the algebraic operations such as addition
and multiplications, etc. are well defined. In one variable Calculus one does not realize how conveniently a
point or an input to the function is a real number and as such they are conveniently open to algebra and
possess convenient Algebraic properties. Notions such as addition, subtraction and magnification, which are
naturally defined on the real numbers, are all absent when we deal with the points in the plane or with the
n-tuples which are to be inputs to the multi-variate functions. Therefore, f((x, y)+(a, b)) or f(k(x, y)) are
not meaningful. To deal this problem we (the textbook) decided to work with R2instead of the plane, and
working with R3instead of space. In general Rnis considered to be the collection of n-tuples with additional
Algebraic properties (please read page 4 very carefully and see how the textbook defines addition and scalar
multiplication for n-tuples.) The advantage of working with R2,R3or Rnis that they are structures which
embed the plane or three dimensional space etc, yet they consistently embody algebraic properties; the
elements of these spaces are known as vectors. (Note: here, the term consistently means all the theory of
Linear Algebra!)
In section 1-1 we attempt at generalizing the elementary concepts such as magnitude and distance to
Rn. The concept of norm (see the linear algebra notes,) conveniently captures whatever the concept of
magnitude could mean. In order to define the norm we appeal to the dot product. While Rnis just a vector
space, without the dot product or the norm, the addition of dot product and the Euclidean norm (it follows
Pythagorean identity) to this vector space makes the Rninto Euclidean space. In this space we have addition
of vectors, scalar multiplication of vectors and real numbers (scalars), the dot product, and the norm all
defined, and this makes the space Rnseemingly richer than R. However throughout the course we will be
using Ras model and we will build Rnsimilar to the properties of R.
Triangle and Cauchy inequalities are generalizations of similar ideas from R(see Deja Vu notes on section
Inequality 1.3 itself deserves a note (to be posted.) And in the space R3we define another product, cross
product. Even though this type of product is strictly defined on R3section 5.9 attemps at introducing certain
objects in Rnknown as the wedge product, which is the generalization of this cross product (applied to higher
spaces.) The wedge product is closely related to the notions of ’determinant’ and this is already in page 7.
Another useful concept introduced in this section is that of a direction in Rn. (See the first example of
this in middle of the proof of Cauchy inequality, while the geometric meaning of this is not intended.) While
in Rthere is two option for a variable, to go forward or backward, to the right or to the left, in Rnthere
is not such restrictions, and a variable can be completely confused in moving about as there are no specific
directions that they are bound to obey. This is a serious problem for the space Rn. Much of the differential
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