(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}.