Note

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.

Definition: Column Vector

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 $a$.

Definition: Real Vector Space

$\R^n$ is the set of column $n$-vectors with real components.

Definition: Transpose

The transpose of a given column vector $a$ is a row vector with corresponding elements, denoted

$$\transpose{ \b{ a }} = (a_1, a_2, …, a_n)$$

$$\b{ a } + \b{ b } = \transpose{ [ a_1 + b_1, a_2 + b_2, …, a_{ n } + b_n] }$$

Exist a unique zero vector.

For each vector $\b{ a }$ exist $- \b{ a }$.

Definition: Scalar Multiplication

$c \b{ a } = \transpose{(ca_1, ca_2, …, ca_{ n })}$

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: $-1 \b{ a } = - \b{ a }$.

Definition: Linearly Independent

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.

Note

Any set of vectors that contains the zero vector is linearly dependent.

Definition: Linear Combination

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 $\set{ \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k }}$.

Corollary

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.

Definition: Subspace of $\R^n$

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.

Definition: Span

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,

$$\set{ \sum_{ i=1 }^{ k } c_i \b{ a }_i \mid \b{ a }_1, \b{ a }_2, …, \b{ a }_{ k } \in \R }$$

Definition: Basis

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 $V$.

Corollary

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

$$\b{ a } = c_1 \b{ a }_1 + c_2 \b{ a }_2 + … + c_{ k } \b{ a }_{ k }$$