A transformation (or function/map) is a rule that assigns every element from one set to an element in the same or another set. Formally, if we have two sets $X$ and $Y$, a transformation is a function:

$$T:X\to Y,$$

which means that for each element $x\in X$, there is a unique corresponding element $T(x)\in Y$.

Linear Transformations

Let $V$ and $W$ be vector spaces over the same field. A function

$$T:V\to W$$

is called a linear transformation if for all vectors $\mathbf{u},\mathbf{v}\in V$ and every scalar $c$, the following two properties hold:

1. Additivity:

$$T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$$

1. Homogeneity (Scalar Multiplication):

$$T(c\mathbf{u})=cT(\mathbf{u})$$

Matrices

When dealing with finite-dimensional vector spaces, any linear transformation can be represented by a matrix.
If $T$ is a linear transformation from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$, there exists an $m\times n$ matrix $A$ such that

$$T(\mathbf{x})=A\mathbf{x}$$

for every vector $\mathbf{x}\in {\mathbb{R}}^n$.

---

Matrix-Vector Multiplication Linearity

Given a matrix $A\in {\mathbb{R}}^{m\times n}$ and $\mathbf{x}\in {\mathbb{R}}^n$

$$(A\mathbf{x}{)}_i=\sum _{j=1}^n{a}_{ij}{x}_j,\text{for }i=1,2,\dots ,m.$$

To verify the linearity of the map $\mathbf{x}\mapsto A\mathbf{x}$, we check two properties:

1. Additivity:
   For any vectors $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$, we have:

$$A(\mathbf{x}+\mathbf{y})=A\mathbf{x}+A\mathbf{y}.$$

1. Homogeneity:
   For any vector $\mathbf{x}\in {\mathbb{R}}^n$ and any scalar $c$, we have:

$$A(c\mathbf{x})=c(A\mathbf{x}).$$

A direct computation shows that these properties hold due to the distributive and scalar multiplication properties inherent in the definition of matrix multiplication.

But are all linear transformations Matrices?

To prove that every linear transformation can be written as a matrix, we must also show:

• Matrix Representation Existence:
  For a given linear transformation

$$T:V\to W,$$

where $V$ and $W$ are finite-dimensional vector spaces, choose bases

$$\{v_1,\dots ,v_n\}\text{and}\{w_1,\dots ,w_m\},$$

Then express each image

$$T(v_i)=a_{1i}{w}_1+a_{2i}{w}_2+\cdots +a_{mi}{w}_m.$$

These coefficients $a_{ji}$ form the columns of the matrix $A$ such that for any $\mathbf{x}\in V$, the action of $T$ is given by

$$T(\mathbf{x})=A\mathbf{x},$$

when $\mathbf{x}$ is expressed in the chosen basis.

• Uniqueness Relative to the Bases:
  The constructed matrix is unique for the chosen bases, which guarantees a one-to-one correspondence between linear transformations and their matrix representations.

In More detail

Since $T$ is linear, the image of each basis vector $v_i$ can be uniquely written as a linear combination of the basis vectors of $W$. That is, for each $i=1,2,\dots ,n$,

$$T(v_i)=a_{1i}{w}_1+a_{2i}{w}_2+\cdots +a_{mi}{w}_m,$$

where $a_{ji}$ are scalars.
We form the matrix $A$ using these coefficients as columns:

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right).$$

Any vector $\mathbf{x}\in V$ can be written as a linear combination of the basis vectors:

$$\mathbf{x}=x_1{v}_1+x_2{v}_2+\cdots +x_n{v}_n.$$

Applying the linear transformation $T$ gives:

$$T(\mathbf{x})=x_{1}T(v_1)+x_{2}T(v_2)+\cdots +x_{n}T(v_n).$$

Substitute the expressions for $T(v_i)$:

$$T(\mathbf{x})=x_1(a_{11}{w}_1+a_{21}{w}_2+\cdots +a_{m1}{w}_m)+\cdots +x_n(a_{1n}{w}_1+a_{2n}{w}_2+\cdots +a_{mn}{w}_m).$$

Rearrange the sums to group the coefficients for each $w_j$:

$$T(\mathbf{x})=\left(\sum _{i=1}^n{a}_{1i}{x}_i\right)w_1+\left(\sum _{i=1}^n{a}_{2i}{x}_i\right)w_2+\cdots +\left(\sum _{i=1}^n{a}_{mi}{x}_i\right)w_m.$$

This is exactly the result of the matrix multiplication:

$$A\left(\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right).$$

$A\vec{x}$ gives the coordinates in basis of $W$.

Thus, every linear transformation $T:V\to W$ can be represented by a matrix $A$. The columns of $A$ are the coordinate representations of the images of the basis vectors of $V$, and the transformation of any vector is achieved via matrix multiplication.