Linear Maps

homomorphism

Let $𝑼$ and $𝑾$ be vector spaces. A function $𝝋 : 𝑼 \to 𝑾$ is a linear map if

$\forall \vec{𝒖}, \vec{𝒗} \in 𝑼 𝝋 (\vec{𝒖} + \vec{𝒗}) = 𝝋 (\vec{𝒖}) + 𝝋 (\vec{𝒗})$
$\forall α \in 𝕂, \forall \vec{𝒖} \in 𝑼 𝝋 (α \vec{𝒖}) = α 𝝋 (\vec{𝒖})$

homomorphism

The function $𝝋 : 𝑼 \to 𝑾$ is a linear map if and only if

\forall α, β \in 𝕂, \forall \vec{𝒖}, \vec{𝒗} \in 𝑼 𝝋 (α \vec{𝒖} + β \vec{𝒗}) = α 𝝋 (\vec{𝒖}) + β 𝝋 (\vec{𝒗})

isomorphism

The function $𝝋 : 𝑼 \to 𝑾$ is a isomorphism if

$𝝋$ is linear
𝝋 is one-to-one and onto
- $\forall \vec{𝒖}, \vec{𝒗} \in 𝑼 𝝋 (\vec{𝒖}) = 𝝋 (\vec{𝒗}) \Rightarrow \vec{𝒖} = \vec{𝒗}$
- $\forall \vec{𝒘} \in 𝑾 \exists \vec{𝒖} \in 𝑼 : 𝝋 (\vec{𝒖}) = \vec{𝒘}$

Terminology

Let $𝝋 : 𝑼 \to 𝑾$ be a function:

linear: $𝝋 (α \vec{𝒖} + β \vec{𝒗}) = α 𝝋 (\vec{𝒖}) + β 𝝋 (\vec{𝒗})$
isomorphism: one-to-one and onto
transformation: $𝑼 = 𝑾$
automorphism: transformation + isomorphism

Properties

zero vector

Let $𝝋 : 𝑼 \to 𝑾$ be a linear function

𝝋 (\vec{𝒐_{𝑼}}) = \vec{𝒐_{𝑾}}

preserving dependency

Let $𝝋 : 𝑼 \to 𝑾$ be a linear function, then it maps linearly dependent vectors to linearly dependent vectors.

preservation of spanning sets

Let $𝝋 : 𝑼 \to 𝑾$ be a onto linear function, then it maps a spanning set vectors to a spanning set of vectors.

Isomorphic vector spaces

Representation map

Let $𝑽$ be vector space over $𝕂$ and $𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}$ be a basis for $𝑽$ . The representation map

ℛ_{𝑩} : 𝑽 → 𝕂^{d}

is defined as

ℛ_{𝑩} (\vec{𝒖}) = ℛ_{𝑩} (ξ_{1} \vec{𝒃_{1}} + ξ_{2} \vec{𝒃_{2}} + \dots + ξ_{d} \vec{𝒃_{d}}) = (\begin{matrix} ξ_{1} \\ ξ_{2} \\ ⋮ \\ ξ_{d} \end{matrix})

Representation map

The representation map $ℛ_{𝑩} : 𝑽 → 𝕂^{d}$ is an isomorphism.

inverse isomorphism

If $𝝋 : 𝑼 \to 𝑾$ is an isomorphism then $𝝋^{-1} : 𝑾 \to 𝑼$ is also an isomorphism.

equivalence relation

Isomorphism is an equivalence relation.

isomorphism implies dimension equality

If two vector space are isomorphic then they have the same dimension.

dimension equality implies isomorphism

If two vector space have the same dimension then they are isomorphic.

Linear extensions

basis action

A homomorphism is defined by its action on a basis. That is, if

𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}

is a basis for $𝑼$ and

\vec{𝒘_{1}}, \vec{𝒘_{2}}, \dots, \vec{𝒘_{d}} \in 𝑾

not necessarily distinct vectors then there exists a unique homomorphism $𝝋 : 𝑼 \to 𝑾$ such that

𝝋 (\vec{𝒃_{i}}) = \vec{𝒘_{i}}

linear extension

Let $𝑼$ and $𝑾$ be two vector spaces and

𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{d}}}

be a basis for $𝑼$ . Let

\tilde{𝝋} : 𝑩 \to 𝑾

be a function. Then $\tilde{𝝋}$ is extended linearly to

𝝋 : 𝑼 \to 𝑾

𝝋 (\vec{𝒖}) = 𝝋 (ξ_{1} \vec{𝒃_{1}} + ξ_{2} \vec{𝒃_{2}} + \dots + ξ_{d} \vec{𝒃_{d}}) = ξ_{1} \tilde{𝝋} (\vec{𝒃_{1}}) + ξ_{2} \tilde{𝝋} (\vec{𝒃_{2}}) + \dots + ξ_{d} \tilde{𝝋} (\vec{𝒃_{d}})

Range and kernel

subspace preservation

Under a homomorphism, the image of any subspace of the domain is a subspace of the codomain.

Rank

The range space of a homomorphism $𝝋 : 𝑼 \to 𝑾$ is

𝑹 (𝝋) = {𝝋 (\vec{𝒖}) ∣ \vec{𝒖} \in 𝑼}

The dimension of $𝑹 (𝝋)$ is called rank of $𝝋$ .

subspace pre-image

For any homomorphism, the inverse image of a subspace of the codomain is a subspace of the domain.

kernel

The kernel or the null space of a homomorphism $𝝋 : 𝑼 \to 𝑾$ is

𝐤𝐞𝐫 (𝝋) = {\vec{𝒖} \in 𝑼 ∣ 𝝋 (\vec{𝒖}) = \vec{𝒐_{𝑾}}}

The dimension of $𝐤𝐞𝐫 (𝝋)$ is called nullity of $𝝋$ .

rank nullity theorem

Let $𝝋 : 𝑼 \to 𝑾$ be a homomorphism then

\dim (𝑼) = \dim (𝑹 (𝝋)) + \dim (𝐤𝐞𝐫 (𝝋))

Matrix representation

matrix representation 1/2

Let $𝑩 = {\vec{𝒃_{1}}, \vec{𝒃_{2}}, \dots, \vec{𝒃_{m}}}$ be a basis for $𝑼$ and $𝑮 = {\vec{𝒈_{1}}, \vec{𝒈_{2}}, \dots, \vec{𝒈_{n}}}$ be a basis for $𝑾$ . Let $𝝋 : 𝑼 \to 𝑾$ be a homomorphism and

ℛ_{𝑮} (𝝋 (\vec{𝒃_{1}})) = (\begin{matrix} ξ_{1, 1} \\ ξ_{2, 1} \\ ⋮ \\ ξ_{n, 1} \end{matrix}) ℛ_{𝑮} (𝝋 (\vec{𝒃_{2}})) = (\begin{matrix} ξ_{1, 2} \\ ξ_{2, 2} \\ ⋮ \\ ξ_{n, 2} \end{matrix}) \dots ℛ_{𝑮} (𝝋 (\vec{𝒃_{m}})) = (\begin{matrix} ξ_{1, m} \\ ξ_{2, m} \\ ⋮ \\ ξ_{n, m} \end{matrix})

matrix representation 2/2

Then the matrix representation of $𝝋$ from basis $𝑩$ to basis $𝑮$ is

ℛ_{𝑩 \to 𝑮} (𝝋) = {(\begin{matrix} ξ_{1, 1} & ξ_{1, 2} & \dots & ξ_{1,m} \\ ξ_{2, 1} & ξ_{2, 2} & \dots & ξ_{2,m} \\ ⋱ \\ ξ_{n, 1} & ξ_{n, 2} & \dots & ξ_{n, m} \end{matrix})}_{𝑩 \to 𝑮}

Change of basis

change of basis

Let $𝑩$ and $𝑫$ be a bases for $𝑼$ . The change of basis matrix from $𝑩$ to $𝑫$ is

ℛ_{𝑩 \to 𝑫} (𝒊𝒅)

where $𝒊𝒅 : 𝑼 \to 𝑼$ is the identity map on $𝑼 .$

invertible matrices

A matrix is a change of basis matrix if and only if it is invertible.

matrix relations

\begin{matrix} 𝑼_{𝑩} & \to & 𝝋_{𝑩 \to 𝑮} & \to & 𝑾_{𝑮} \\ ↓ & ↓ \\ {𝒊𝒅}_{𝑩 \to 𝑫} & {𝒊𝒅}_{𝑮 \to 𝑬} \\ ↓ & ↓ \\ 𝑼_{𝑫} & \to & 𝝋_{𝑫 \to 𝑬} & \to & 𝑾_{𝑬} \end{matrix}