(1) In R^n, none of n, R, or R^n has been defined. Better: Let n be a positive integer, and let R be the set of real numbers. Then R^n is the set of all n-tu

(1) In R^n, none of n, R, or R^n has been defined. Better: Let n be a positive integer, and let R be the set of real numbers. Then R^n is the set of all n-tuples over the real numbers. Such an n-tuple is a list of n numbers; e.g., (1, 3, 2.6, 1, 7) is an n-tuple for n = 5. For positive integers m and n, an m x n (read "m by n") (real) 'matrix' as a rectangular array of real numbers with m rows and n columns. Typically for integer i = 1, 2, ..., m and integer j = 1, 2, ..., n, in m x n matrix A = [a_{ij}] a_{ij} is the number, 'component', in row i and column j. An example of a 2 x 3 matrix is 1 9 4 4 0 -1 (typically surrounded with square brackets not easy to type without, say, TeX). Typically an n-tuple x in R^n is regarded as an n x 1 matrix. In this way, with m x n matrix A, n x 1 vector b, and matrix multiplication (we have yet to define) Ax, we can write Ax = b as a system of m linear equations in n unknowns x, and this is the most important foundation and application of linear algebra. (2) Similarly for positive integer n and the set of complex numbers C, C^n is the set of all n-tuples of complex numbers. And, we can have an m x n matrix of complex numbers. Once get very far in linear algebra, complex numbers become essential if only from the fundamental theorem of algebra (that is, from factoring a polynomial). In particular, eigenvalues are important, and they are often complex numbers. (3) R^n and C^n are examples of 'vector spaces', but there are also many other examples. For a 'vector space', we have a non-empty set V of 'vectors' and a 'field' F. Note: A 'field' is defined carefully in courses in abstract algebra, but the two leading examples in practice are R and C. Most commonly F = R or F = C, but there is also interest in other fields, e.g., the integers modulo a prime number. Here we assume that F = R or F = C. On V we have an operation 'addition' denoted by +. So, for vectors x in V and y in V, there is a vector x + y in V. Addition is commutative so that x + y = y + x. Vector addition is also associative, that is, for x, y, z in V, (x + y) + z = x + (y + z) Also there is a vector 0 in V so that for any x in V we have 0 + x = x. Also for x in V, there is -x in V so that x + (-x) = x - x = 0. Or set V with operation + forms a commutative group (defined carefully in courses in abstract algebra). On V with F we have an operation 'multiplication'. Then for x in V and a in F, with this 'multiplication' the 'product' ax in V. For x, y in V and a, b in F, we have two distributive laws, (a + b)x = ax + bx and a(x + y) = ax + ay. Also, for 0 in F, 0x is the vector 0. For the vector 0, a0 is also the vector 0. We have that (-1)x = -x. For some examples of vector spaces, (A) let F = R and let V be the set of all monophonic audio signals exactly 10 seconds long, (B) let F = R and let V be the set of all real valued random variables X where E[X^2] is finite, and (C) let F = R and for m x n matrix A and n x 1 vector x and m x 1 vector b, let V be the set of all x such so that Ax = b (so far the matrix product Ax has not been defined, but it will be defined so that Ax = b is a system of m linear equations in n unknowns, the components of x). (3) For set S, x in S means that x is an 'element' of S. (4) The notation (R, R, R) = R^3 is neither good nor standard. The notation R^3 = R X R X R as the set theory Cartesian product is standard. Actually, we already know just what (R, R, R) is -- a 3-tuple and not a set of all 3-tuples of real numbers. (5) R^{m x n} is questionable notation and, anyway, needs a definition. (6) For the "power" of that mathematics, might illustrate that with some applications. One of the most important is solving systems of linear equations. For more, a matrix can describe a linear transformation, and linearity is a powerful assumption that commonly holds with good accuracy in practice. So, vectors and matricies become powerful means of working with linear systems where a matrix describes the system and the inputs and outputs of that system are vectors. From G. F. Simmons, the two pillars of analysis in mathematics are continuity and linearity, and linear algebra emphsizes both, especially linearity. More examples of linearity include both differentiation and integration in calculus and the Fourier transform. (6) The definition of inverse function is wrong. That definition holds only when the given function f is 1-1. Even if f is not 1-1, the inverse function is still useful, but its values can be a set of values, that is, the inverse function is multiply valued. E.g., just take f(x) = x^2. Then the f^{-1}(1) = {-1, 1}.