Chapter 4 - Linear Spaces
4.13 Basic Concepts
Results
Definition 1
A nonempty set $L$ of elements $x,y,z,\ldots$ is said to be a
linear space (or
vector space)
if it satisfies the following three axioms.
(1) Any two elements $x,y\in L$ uniquely determine a third element $x+y\in L$, called the
sum
of $x$ and $y$, such that
(a) $x + y = y + x$ (commutativity)
(b) $(x + y) + z = x + (y + z)$ (associativity)
(c) There exists an element $0\in L$, called the zero element with the property that
$x + 0 = x$ for every $x\in L$
(d) For every element $x\in L$ there exists an element $-x\in L$, called the negative of $x$
with the property that $x + (-x) = 0$.
(2) Any number $\alpha$ and any element $x\in L$ uniquely determines an element $\alpha x\in L$, called the
product of $\alpha$ and $x$, such that
(a) $\alpha(\beta x) = (\alpha\beta)x$
(b) $1\cdot x = x$
(3) The operations of addition and multiplication obey the two distributive laws.
(a) $(\alpha + \beta)x = \alpha x + \beta x$
(b) $\alpha(x + y) = \alpha x + \alpha y$
The points $x$, $y$ are called points or vectors, and $\alpha$, $\beta$ are called scalars.
Definition 2
Two linear spaces $L$ and $L^*$ are said to be
isomorphic if there is a one-to-one
correspondence $x\leftrightarrow x^*$ between $L$ and $L^*$ which preserves operations,
in the sense that
$$
x\leftrightarrow x^*
,\quad
y\leftrightarrow y^*
$$
where $x,y\in L$ and $x^*,y^*\in L^*$, implies
$$
x + y\leftrightarrow x^* + y^*
$$
and
$$
\alpha x \leftrightarrow \alpha x^*
$$
for some arbitrary $\alpha$.
Linear Dependence
The elements $x,y,\ldots,w$ of a linear space $L$ are
linearly dependent if there exists numbers $\alpha, \beta,\ldots,\lambda$,
not all zeros, such that
$$
\alpha x + \beta y + \ldots + \lambda w = 0.
$$
If no such numbers exists, the points are said to be
linearly independent. In an $n$-dimensional linear space, any set of
$n$ linearly independent elements if called a
basis. Finite $n$-dimensional spaces are covered in linear algebra, while
real analysis is more about the study of infinite dimensional spaces.
Subspace
Given a nonempty subset $L'$ of a linear space $L$, if $L'$ satisfies all the axioms of a linear space, then $L'$ is said to
be a
subspace of $L$.
Factor spaces
Let $L$ be a linear space and $L'$ a subspace of $L$. The elements $x,y\in L$ belong to the same
(residue) class generated by $L'$ if the difference $x-y\in L'$. The set of all such classes
is called the
factor space or
quotient space of $L$ relative to $L'$, denoted by $L|L'$.
The dimension of the factor space $L|L'$ is called the
codimension of $L'$ in $L$.
Theorem 1
Every factor space $L|L'$ is a linear space.
Theorem 2
Let $L'$ be a subspace of a linear space $L$. Then $L'$ has a finite codimension $n$ if and only if there are
linearly independent elements $x_1,\ldots, x_n$ in $L$ such that every element $x\in L$ has a unique
representation of the form
$$
x = \alpha_1x_1 + \ldots + \alpha_nx_n + y,
$$
where $\alpha_1,\ldots,\alpha_n$ are numbers and $y\in L'$.
Linear functionals
A numeric function $f:L\rightarrow\R$ is called a
functional on $L$. (A little different from the use in 12.1).
A functional $f$ is said to be
additive if,
$$
f(x + y) = f(x) + f(y)
$$
for all $x,y\in L$, and
homogeneous if
$$
f(\alpha x) = \alpha f(x)
$$
for every number $\alpha$. A functional defined on a
complex linear space is called
conjugate-homogeneous if,
$$
f(\alpha x) = \overline{\alpha}f(x)
$$
for any $\alpha\in\mathbb{C}$, where $\overline{\alpha}$is the complex conjugate of $\alpha$.
Null Space
Let $f$ be a functional defined on a linear space $L$. Then the set $L_f$ of all elements $x\in L$ such that
$$
f(x) = 0
$$
is called the
null space of $f$, where $f$ is assumed to be
nontrivial, i.e. $f\not= 0$.
Theorem 3
Let $x_0$ be any fixed element of $L - L_f$. Then every element $x\in L$ has a unique representation of the form
$$
x = \alpha x_0 + y
$$
where $y\in L_f$.
Corollary 1
Two elements $x_1$ and $x_2$ belong to the same class generated by $L_f$ if and only if $f(x_1) = f(x_2)$.
Corollary 2
$L_f$ has codimension 1.
Corollary 3
Two nontrivial linear functionals $f$ and $g$ with the same null space are proportional.
Hyperplane
Given a linear space $L$, let $L'\subset L$ be any subspace of codimension 1. Then every class in $L$
generated by $L'$ is called a
hyperplane "parallel to $L$". In other words, a hyperplane $M'$
parallel to a subspace $L'$ is the set obtained by subjecting $L'$ to the parallel displacement determined
by the vector $x_0\in L$, so that
$$
M' = L' + x_0 = \{x\mid x = x_0 + y, y\in L'\}.
$$
(Closely related to affine spaces).
Theorem 4
Given a linear space $L$, let $f$ be a nontrivial linear functional on $L$. Then the set
$$
M_f = \{x\mid f(x) = 1\}
$$
is a hyperplane parallel to the null space $L_f$ of the functional. Conversely, let $M' = L' + x_0$
where $x_0\not\in L'$, be any hyperplane parallel to a subspace $L'\subset L$ of codimension 1 and not
passing through the origin. Then there exists a unique linear functional $f$ on $L$ such that
$$
M' = \{x\mid f(x) = 1\}.
$$
Problem 1
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Problem 2
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Problem 3
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Problem 4
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Problem 5
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Problem 6
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