We do not discuss everything in great detail here since everything has been discussed in MATH 115A Section 1.2. This note merely serves as a quick review.
A column $n$-vector is an array of $n $ numbers, denoted
$$\b{ a } = {\Vcr{ a_1 }{ a_2 }{ \vdots }{ a_n }} $$
$a_i $ is called the $i $th component of the vector $a$.
$\R^n $ is the set of column $n $-vectors with real components.
The transpose of a given column vector $a $ is a row vector with corresponding elements, denoted
$$\transpose{ \b{ a }} = (a_1, a_2, …, a_n)$$
$$\b{ a } + \b{ b } = \transpose{ [ a_1 + b_1, a_2 + b_2, …, a_{ n } + b_n] } $$
Vector addition follows: commutativity, associativity.
Exist a unique zero vector.
For each vector $\b{ a } $ exist $- \b{ a } $.
$c \b{ a } = \transpose{(ca_1, ca_2, …, ca_{ n })}$
Scalar multiplication follows: distributive law, associativity.
The scalar $1$ satisfies: $1 \b{ a } = \b{ a } $.
Any scalar $c$ satisfies: $c \b{ 0 } = \b{ 0 } $.
The scalar $0 $ satisfies: $0 \b{ a } = 0 $.
The scalar $-1 $ satisfies: $-1 \b{ a } = - \b{ a } $.
if $c_1 \b{ a }_1 + c_2 \b{ a }_2 + … + c_{ k } \b{ a }_{ k } = 0 \implies \forall i = 1, …, k, c_i = 0$, then
$\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }}$ is linearly independent. Otherwise, the set is linearly dependent.
Any set of vectors that contains the zero vector is linearly dependent.
if $\exists c_1, c_2, …, c_{ k }$ such that $\b{ a } = c_1 \b{ a }_1 + c_2 \b{ a }_2 + … + c_{ k } \b{ a }_{ k } $, then $\b{ a } $ is a linear combination of $\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }} $.
A set of vectors $\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }}$ is linearly dependent $\iff$ $a_i, i \in \set{1, …, k} $ is a linear combination of the remaining vectors.
A subset $W $ of $\R^n $ is called a subspace of $\R^n $ if $W $ is closed under the operation of vector addition and scalar multiplication.
Let $\b{ a }_1, \b{ a }_2, …, \b{ a }_{ k } $ be arbitrary vectors in $\R^n $. The set of all their linear combinations is called the span of $\b{ a }_1, \b{ a }_2, …, \b{ a }_{ k } $ and is denoted
$$\spa{ \set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }}} $$
or more literally,
$$\set{ \sum_{ i=1 }^{ k } c_i \b{ a }_i \mid \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k } \in \R } $$
Given a vector space $V $, any set of linearly independent vectors $\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }} \subseteq V $ s.t. $V = \spa{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }} $ is referred to as a basis of the vector space $V $.
If $\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }} $ is a basis of a vector space $V $, then any vector $\b{ a } $ of $V $ can be represented uniquely as
$$\b{ a } = c_1 \b{ a }_1 + c_2 \b{ a }_2 + … + c_{ k } \b{ a }_{ k } $$