Linear Mappings

Table of contents
  1. Linear Transformation
    1. Composite Linear Transformation
    2. Transformation Matrix
  2. Types of Linear Mappings
    1. Homomorphism
    2. Isomorphism
      1. When are two vector spaces isomorphic?
      2. Inverse Mapping of an Isomorphism
    3. Endomorphism
    4. Automorphism
      1. Identity Mapping
  3. Change of Basis of Linear Transformation
    1. Matrix Equivalence
    2. Matrix Similarity
    3. Invariance Under Basis Change
  4. Bi-Linear Mapping
    1. Symmetric Bi-Linear Mapping
    2. Positive Definite Bi-Linear Mapping

Linear Transformation

Linear transformation or linear mapping or vector space homomorphism is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication and closure.

This can be summarized as:

For vector spaces $V$, $W$, a mapping $\Phi: V \rightarrow W$ is a linear transformation if $\forall \mathbf{x}, \mathbf{y} \in V$ and $\forall \lambda, \psi \in \mathbb{R}$:

$$ \Phi(\lambda \mathbf{x} + \psi \mathbf{y}) = \lambda \Phi(\mathbf{x}) + \psi \Phi(\mathbf{y}) $$

Composite Linear Transformation

Composite linear transformation is also a linear transformation.

If $\Phi: V \rightarrow W$ and $\Psi: W \rightarrow X$ are linear transformations, then $\Psi \circ \Phi$ is also a linear transformation.

Transformation Matrix

With respect to ordered bases.

Linear transformations can be represented by matrices.

Let $V$ and $W$ be vector spaces with ordered bases $B = (\mathbf{b}_1, \dots, \mathbf{b}_n)$ and $C = (\mathbf{c}_1, \dots, \mathbf{c}_m)$ respectively.

A linear mapping $\Phi: V \rightarrow W$ of

\[\Phi(\mathbf{b}_j) = \sum_{i=1}^m a_{ij} \mathbf{c}_i\]

is represented by the matrix

\[A_{\Phi} = \begin{bmatrix} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \dots & a_{mn} \end{bmatrix}\]

or for short

\[A_{\Phi}(i, j) = a_{ij}\]

$A_{\Phi}$ is called the transformation matrix of $\Phi$ with respect to the ordered bases.


Types of Linear Mappings

Review the general concepts of injective, surjective, bijective mappings.

Homomorphism

Let $\Phi: V \rightarrow W$ be a linear transformation.

$\Phi$ is a homomorphism.

Isomorphism

Let $\Phi: V \rightarrow W$ be a linear transformation.

$\Phi$ is an isomorphism if it is bijective.

When are two vector spaces isomorphic?

Finite-dimensional vector spaces $V$ and $W$ are isomorphic if and only if they have the same dimension.

$$ V \cong W \iff \dim(V) = \dim(W) $$

More details

There exists a bijection $\Phi: V \rightarrow W$ when $\dim(V) = \dim(W)$.

This means vector spaces of the same dimension are essentially the same thing, only represented differently.

Inverse Mapping of an Isomorphism

The inverse mapping of an isomorphism $\Phi^{-1}: W \rightarrow V$ is also an isomorphism, by the bijectivity of the mapping.

Endomorphism

Let $\Phi: V \rightarrow V$ be a linear transformation.

$\Phi$ is an endomorphism.

Automorphism

Let $\Phi: V \rightarrow V$ be a linear transformation.

$\Phi$ is an automorphism if it is bijective.

Identity Mapping

An identity mapping $\mathrm{id}_V: V \rightarrow V,\, x \mapsto x$ is an automorphism in $V$.


Change of Basis of Linear Transformation

Let $\Phi: V \rightarrow W$ be a linear transformation.

Let $B = (\boldsymbol{b}_1, \dots, \boldsymbol{b}_n)$ and $\tilde{B} = (\tilde{\boldsymbol{b}}_1, \dots, \tilde{\boldsymbol{b}}_n)$ be ordered bases of $V$.

Let $\boldsymbol{S}$ be the transformation matrix of $\mathrm{id}_V$ which maps coordinates of $\tilde{B}$ to $B$.

Let $C = (\boldsymbol{c}_1, \dots, \boldsymbol{c}_m)$ and $\tilde{C} = (\tilde{\boldsymbol{c}}_1, \dots, \tilde{\boldsymbol{c}}_m)$ be ordered bases of $W$.

Let $\boldsymbol{T}$ be the transformation matrix of $\mathrm{id}_W$ which maps coordinates of $\tilde{C}$ to $C$.

Change of basis matrices $\boldsymbol{S}$ and $\boldsymbol{T}$ are invertible because they are automorphisms.

Change of Basis of Linear Transformation

If $A_{\Phi}$ is the transformation matrix of $\Phi$ with respect to $B$ and $C$, and $\tilde{A}_{\Phi}$ is the transformation matrix of $\Phi$ with respect to $\tilde{B}$ and $\tilde{C}$, then

$$ \tilde{A}_{\Phi} = T^{-1} A_{\Phi} S $$

Matrix Equivalence

We say that $A$ and $\tilde{A}$ are equivalent if there exist invertible matrices $S$ and $T$ such that

\[\tilde{A} = T^{-1} A S\]

Matrix Similarity

If $A$ and $\tilde{A}$ are square matrices and there exists an invertible matrix $S$ such that

\[\tilde{A} = S^{-1} A S\]

then we say that $A$ and $\tilde{A}$ are similar.

Similar, in that transformation $A$ and $\tilde{A}$ are essentially the same, except for the change of basis represented by $S$.

Invariance Under Basis Change

Some characteristic values of a linear transformation (or matrix) do not change under basis change.

Some of these are:


Bi-Linear Mapping

As seen above, linear mapping means it is a mapping closed under vector addition and scalar multiplication.

Bi-linear mapping is similar, just that this mapping takes two arguments instead of one.

For vector spaces $V$, $W$, $X$, a mapping $\Omega: V \times W \rightarrow X$ is a bi-linear mapping if $\forall \mathbf{x} \in V, \forall \mathbf{y} \in W, \forall \mathbf{z} \in W$ and $\forall \lambda, \psi \in \mathbb{R}$:

$$ \begin{gather*} \Omega(\lambda \mathbf{x} + \psi \mathbf{y}, \mathbf{z}) = \lambda \Omega(\mathbf{x}, \mathbf{z}) + \psi \Omega(\mathbf{y}, \mathbf{z}) \\[1em] \Omega(\mathbf{x}, \lambda \mathbf{y} + \psi \mathbf{z}) = \lambda \Omega(\mathbf{x}, \mathbf{y}) + \psi \Omega(\mathbf{x}, \mathbf{z}) \end{gather*} $$

More generally \[\begin{align*} \Omega\left(\sum_{i=1}^n \lambda_i \mathbf{x}_i, \sum_{j=1}^m \psi_j \mathbf{y}_j\right) &= \lambda_1 \psi_1 \Omega(\mathbf{x}_1, \mathbf{y}_1) + \dots + \lambda_2 \psi_1 \Omega(\mathbf{x}_2, \mathbf{y}_1) + \dots + \lambda_n \psi_m \Omega(\mathbf{x}_n, \mathbf{y}_m) \\[1em] &= \sum_{i=1}^n \sum_{j=1}^m \lambda_i \psi_j \Omega(\mathbf{x}_i, \mathbf{y}_j) \end{align*}\]

Symmetric Bi-Linear Mapping

$\Omega$ is symmetric if $V = W$ and $\forall \mathbf{x}, \mathbf{y} \in V$:

$$ \Omega(\mathbf{x}, \mathbf{y}) = \Omega(\mathbf{y}, \mathbf{x}) $$

Positive Definite Bi-Linear Mapping

$\Omega$ is positive definite if $V = W$ and $\forall \mathbf{x} \in V$:

  • When $\mathbf{x} \neq \mathbf{0}$, $\Omega(\mathbf{x}, \mathbf{x}) > 0$.
  • When $\mathbf{x} = \mathbf{0}$, $\Omega(\mathbf{x}, \mathbf{x}) = 0$.