Vector spaces

Vector space

A vector space over $𝕂$ is a non-empty set $𝑽$ of elements called vectors along with two operations vector addition and scalar multiplication

\oplus : 𝑽 \times 𝑽 \to 𝑽 ⊙ : 𝕂 \times 𝑽 \to 𝑽

such that

»	$\vec{𝒗} \oplus \vec{𝒖} = \vec{𝒖} \oplus \vec{𝒗}$	»	$1 ⊙ \vec{𝒗} = \vec{𝒗}$
»	$(\vec{𝒗} \oplus \vec{𝒖}) \oplus \vec{𝒘} = \vec{𝒗} \oplus (\vec{𝒖} \oplus \vec{𝒘})$	»	$α ⊙ (β ⊙ \vec{𝒗}) = (α β) ⊙ \vec{𝒗}$
»	$\exists! \vec{𝒐} \in 𝑽 : \forall \vec{𝒗} \in 𝑽 \vec{𝒐} \oplus \vec{𝒗} = \vec{𝒗}$	»	$α ⊙ (\vec{𝒗} \oplus \vec{𝒖}) = (α ⊙ \vec{𝒗}) \oplus (α ⊙ \vec{𝒖})$
»	$\forall \vec{𝒗} \in 𝑽, \exists! \vec{-𝒗} \in 𝑽 : \vec{𝒗} \oplus \vec{-𝒗} = \vec{𝒐}$	»	$(α + β) ⊙ \vec{𝒗} = (α ⊙ \vec{𝒗}) \oplus (β ⊙ \vec{𝒗})$

Examples

Vectors from geometry
Polynomials of degree at most two
complex numbers over complex numbers

Counterexamples

Vectors with integer components
Polynomials that evaluate to one at two
real numbers over complex numbers

Crazy Vector Space

𝑪𝑽𝑺 = {[\begin{matrix} x \\ y \end{matrix}] ∣ x, y \in ℝ} / ℝ

Operations

\begin{matrix} [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] \oplus [\begin{matrix} x_{2} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{1} + x_{2} - 2 \\ y_{1} + y_{2} \end{matrix}] \\ α ⊙ [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} α x - 2 α + 2 \\ α y \end{matrix}] \end{matrix}

Multiplication with zero

\forall \vec{𝒗} \in 𝑽 0 ⊙ \vec{𝒗} = \vec{𝒐}

Multiplication with

negative one

\forall \vec{𝒗} \in 𝑽 (-1) ⊙ \vec{𝒗} = \vec{-𝒗}

Scalar multiple of zero vector

\forall α \in 𝕂 α ⊙ \vec{𝒐} = \vec{𝒐}

Linear combinations

Linear combination

A vector $\vec{𝒖}$ is a linear combination of

\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{k}}

if there are constants

α_{1}, α_{2}, \dots, α_{k}

such that

\vec{𝒖} = α_{1} \vec{𝒘_{1}} + α_{2} \vec{𝒘_{2}} + \dots + α_{k} \vec{𝒘_{k}}

Linear combination of subsets

If $\vec{𝒖}$ is a linear combination of a subset of $\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{k}}$ then it is a linear combination of all the vectors $\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{k}}$ .

transitivity

Suppose

$\vec{𝒖}$ is a linear combination of $\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{k}}$ and
each $\vec{𝒘_{j}}$ is a linear combination of $\vec{𝒗_{1}}, \vec{𝒗_{2}}, \dots, \vec{𝒗_{t}}$

then $\vec{𝒖}$ is a linear combination of $\vec{𝒗_{1}}, \vec{𝒗_{2}}, \dots, \vec{𝒗_{t}}$ .

matrix multiplication

Let $C = A B$ then

rows of $C$ are linear combinations of the rows of $B$
columns of $C$ are linear combinations of the columns of $A$

Linear (in)dependence

Let ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ be a set of vectors

α_{1} \vec{𝒖_{1}} + α_{2} \vec{𝒖_{2}} + \dots + α_{k} \vec{𝒖_{k}} = \vec{𝒐} \Rightarrow α_{1} = 0, α_{2} = 0, \dots, α_{k} = 0

then the vectors ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ are linearly independent, otherwise they are linearly dependent.

Standard basis

The standard basis vectors are linearly independent i.e., the columns and rows of $𝑰$ are linearly independent.

One vector case

The set ${\vec{𝒖_{1}}}$ is linearly indepedent if and only if $\vec{𝒖_{1}} = \vec{𝒐}$ .

Sets with zero vector

Let ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ be a set of vectors such that there is $i$ such that

\vec{𝒖_{i}} = \vec{𝒐}

then the set is linearly depedent.

Sets with equal vectors

Let ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ be a set of vectors such that there are $i \neq j$ such that

\vec{𝒖_{i}} = \vec{𝒖_{j}}

then the set is linearly depedent.

Linear dependence and combinations

Let ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ be a set of linearly dependent vectors with $2 \leq k$ then there is $t$ such that $\vec{𝒖_{t}}$ is a linear combination of the remaining vectors in the set.

Main Lemma

Let ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{m}}}$ and ${\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{n}}}$ be two non-empty sets of vectors such that

\begin{matrix} \vec{𝒖_{1}} & = & γ_{1, 1} \vec{𝒘_{1}} + γ_{1, 2} \vec{𝒘_{2}} + \dots + γ_{1, n} \vec{𝒘_{n}} \\ \vec{𝒖_{2}} & = & γ_{2, 1} \vec{𝒘_{1}} + γ_{2, 2} \vec{𝒘_{2}} + \dots + γ_{2, n} \vec{𝒘_{n}} \\ ⋮ \\ \vec{𝒖_{m}} & = & γ_{m, 1} \vec{𝒘_{1}} + γ_{m, 2} \vec{𝒘_{2}} + \dots + γ_{m, n} \vec{𝒘_{n}} \end{matrix}

if $m < n$ then ${\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{m}}}$ are linearly dependent.

Square matrices

A square matrix of order $n$ has linearly dependent rows if and only if it has linearly dependent rows.

Subspaces

subspace

Let $𝑽$ be a vector space and $𝑼 \subseteq 𝑽$ . If $𝑼$ is a vector space the same operations then $𝑼$ is a subspace of $𝑽$ .

trivial subspaces

Let $𝑽$ be a vector space, then $𝑽$ is a subspace of $𝑽$ and ${\vec{𝒐}}$ is a subspace of $𝑽$ .

subspace condition

$𝑼$ is a subspace of $𝑽$ if and only if for all $\vec{𝒖}, \vec{𝒘} \in 𝑼$ and for all $α, β \in 𝕂$

α \vec{𝒖} + β \vec{𝒘} \in 𝑼

Span

Let $𝑺 = {\vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}}}$ be a set of vectors. The set of all linear combinations of these vectors is the span of $𝑺$ and denoted by $⟨ 𝑺 ⟩$ .

⟨ 𝑺 ⟩ = ⟨ \vec{𝒖_{1}}, \vec{𝒖_{2}}, \dots, \vec{𝒖_{k}} ⟩

if the set is empty

⟨ \emptyset ⟩ = {\vec{𝒐}}

cosistency condition

A system of linear equation $A \vec{x} = \vec{b}$ has a solution if and only if $\vec{b}$ is in the span of the columns of $A$ .

expanding spans

$⟨ 𝑺 \cup \vec{𝒖} ⟩ = ⟨ 𝑺 ⟩$ if and only if $\vec{𝒖} \in 𝑺$ .

spans are vector spaces

The span of a set of vectors is a vector space.

Basis and dimension

Basis

The set $𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}$ is a basis for a vector space $𝑽$ if

$⟨ 𝑩 ⟩ = 𝑽$
$𝑩$ is linearly independent

Size

Let $𝑩_{1}$ and $𝑩_{2}$ be bases for a vector space $𝑽$

| 𝑩_{1} | = | 𝑩_{2} |

dimension

Let $𝑩$ be basis for a vector space $𝑽$ . The size of $𝑩$ is called the dimension of $𝑽$ .

finite dimensions

A vector space is called finite dimensional it its basis has a finite number of vectors.

extending to a basis

Any linearly independent set can be extended to a basis.

diminishing to a basis

Any spanning set contains a basis.

Coordinates

uniqueness

Let ${\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}$ be a linearly independent set. If

$\vec{𝒖} = α_{1} \vec{𝒃_{1}} + α_{2} \vec{𝒃_{2}} + \dots + α_{d} \vec{𝒃_{d}}$
$\vec{𝒖} = β_{1} \vec{𝒃_{1}} + β_{2} \vec{𝒃_{2}} + \dots + β_{d} \vec{𝒃_{d}}$

Then

α_{1} = β_{1}, α_{2} = β_{1}, \dots, α_{d} = β_{d}

coordinates

Let $\vec{𝒖} \in 𝑽$ , $𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}$ be basis for $𝑽$ and

\vec{𝒖} = ξ_{1} \vec{𝒃_{1}} + ξ_{2} \vec{𝒃_{2}} + \dots + ξ_{d} \vec{𝒃_{d}}

Then $ξ_{1}, ξ_{2}, \dots, ξ_{d}$ are coordinates of $\vec{𝒖}$ with respect to basis $𝑩$ .

Basis description

In a vector space $𝑽$ a subset $𝑩$ is a basis for $𝑽$ if and only if every vector in $𝑽$ can be represented in a unique was as a linear combination of the vectors in $𝑩$ .

Matrix Rank

independent columns

Let $𝐴$ be a square matrix such that there is matrix $𝐵$ such that

𝐴 ⁢ 𝐵 = 𝐼

Then the columns of $𝐴$ are linearly independent.

rank

The rank of a matrix $𝐴$ is the number of linearly independent columns of the matrix and is denoted by

rank (𝐴)

independent rows and columns

Let $𝐴$ be a $m \times n$ matrix. The number of linearly independent rows equals the number of linearly independent columns.

product of elementary matrices

Let $𝐴$ be a square matrix that has linearly independent rows. Then $𝐴$ can be written as a product of elementary matrices.

consistency condition

A system of linear equation $𝐴 \vec{𝒙} = 𝑏$ is consistent if and only if

rank (𝐴) = rank (𝐴 ∣ 𝑏)

dual condition

A system of linear equations $𝐴 \vec{𝒙} = 𝑏$ is consistent if and only if

𝐴^{T} 𝑦 = \vec{𝒐} \Rightarrow 𝑏^{T} 𝑦 = 0