1. Linear transformation:

Let M and N be vector spaces over a field R. Then a function T:M→N is called a linear transformation if, for x, y∈M and α∈R
(i) T(x+y)=T(x)+T(y)
(ii) T(αx)=αT(x)

1.1. Identity transformation:

Let M be a vector space over a field R. Then the function T:M→M defined as T(x)=x, where x∈M, is linear and is known as identity transformation.

1.2. Zero transformation:

Let M and N be vector spaces over a field R. Then the function T:M→N defined as T(x)=0, where x∈M, is linear and is known as zero transformation.

1.3. Examples:

a). Let T: R²→R² such that T(x, y)=(x, -y), where x, y∈R. Then T is linear.

  Proof:

  Let x=(x₁, x₂)and y=(y₁, y₂)∈R². Then T(x₁, x₂)=(x₁, -x₂) and T(y₁, y₂)=(y₁, -y₂). Notice that
  (i)T(x+y)=T[(x₁, x₂)+(y₁, y₂)]=T(x₁+y₁, x₂+y₂)=(x₁+y₁, -( x₂+y₂)) and T(x)+T(y)=(x₁, -x₂)+ (y₁, -y₂)= (x₁+y₁, -( x₂+y₂).
  (ii) T(αx)=T(αx₁, αx₂)=(αx₁, -αx₂) and αT(x)=α(x₁, -x₂)=(αx₁, -αx₂).

b). Let T:R²→R² such that T(x, y)=(x, 0), where x, y∈R. Then T is linear.

  Proof:

  Let x=(x₁, x₂) and y=(y₁, y₂)∈R². Then T(x₁, x₂)=(x₁, -x₂) and T(y₁, y₂)=(y₁, -y₂). Notice that
 (i) T(x+y)=T[(x₁, x₂)+(y₁, y₂)]=T(x₁+y₁, x₂+y₂)=(x₁+y₁, 0) and T(x)+T(y)=(x₁, 0)+ (y₁, 0)=(x₁+y₁, 0).
 (ii) T(αx)=T(αx₁, αx₂)=(αx₁, 0) and αT(x)=α(x₁, 0)=(αx₁, 0).

c). Let T:R²→R² such that T(x, y)=(x, 0), where x, y∈R. Then T is linear.

  Proof:

  Let x=(x₁, x₂) and y=(y₁, y₂)∈R². Then T(x₁, x₂)=(x₁, -x₂) and T(y₁, y₂)=(y₁, -y₂). Notice that
 (i) T(x+y)=T[(x₁, x₂)+(y₁, y₂)]=T(x₁+y₁, x₂+y₂)=( x₂+y₂, x₁+y₁) and T(x)+T(y)=(-x₂, x₁)+ (-y₂, y₁)= ( x₂+y₂, x₁+y₁).
 (ii) T(αx)=T(αx₁, αx₂)=(αx₁, -αx₂) and αT(x)=α(x₂, x₁)=(αx₁, -αx₂).

1.4. Proposition:

Let T:M→N be a linear transformation. Then
(i) T(0)=0; where 0 on the L.H.S is the zero vector of M and 0 on the R.H.S is the zero vector of N.
(ii) T(-x)=-T(x), for all x∈M
(iii) T(x-y)=T(x)-T(y), for all x, y∈M
(iv) T(αx+βy)=αT(x)+βT(y), for all x, y∈M and α, β∈R

1.5. Proposition:

Let T:V→W be a linear map and V and W be finite dimensional vector spaces over R and
B={e₁, e₂, …, e_n} is a basis for V and let x= r₁f₁+r₂ e₂+ ... +r_ne_n, where x∈V, r₁, r₂, …, r_n∈R. Then T(e₁), T(e₂), …, T(e_n) define T, by T(x)= rT(e₁)+sT(e₂)+ … +tT(e_n) .

2. Matrix associated with a linear transformation:

Let T:R³→R² be a linear transformation. Let B₁={f₁, f₂, f₃} and B₂={e₁, e₂} be basis of R³ and R² respectively. Then T(f₁)=ae₁+be₂, T(f₂)=ce₁+de₂ and T(f₃)=ge₁+he₂, for some a, b, c, d, g, h∈R.

Let A=

$$\begin{pmatrix}a & c & e \\\ b & d & f\end{pmatrix}$$

Then matrix A is of order 2×3 and is called the matrix representation of T w.r.t. the basis B₁ and B₂.
In a similar way, one can check that a matrix representation of T:Rⁿ→R^m is of order m×n.

2.1. Example:

Let T:R²→R² be a linear map defined as T(x, y)=(x, -y), where x, y∈R. Then find the matrix associated with the transformation w.r.t. the basis B₁={(1, 0), (0,1)}and B₂={(0, -1), (-1, 0)}.
Let e₁=(1, 0), e₂=(0, 1), f₁=(0, -1) and f₂=(-1, 0). Thus
T(1, 0)=(1, 0)=0.(0, -1)+(-1).(-1, 0) T(0, 1)=(0, -1)= 1.(0, -1)+0.(-1, 0) and hence the matrix representation of T w.r.t. the basis B₁ and B₂ is

3. Linear transformation associated with a matrix:

Let A=

$$\begin{pmatrix}a & c & e \\\ b & d & f\end{pmatrix}$$

be a matrix of order 2×3. Then the linear transformation T:R³→R² associated with the matrix w.r.t. the basis B₁=>={f₁, f₂, f₃} and B₂= B₂={e₁, e₂} of R³ and R²respectively can be obtained as follows:
Define T(f₁)=ae₁+be₂, T(f₂)=ce₁+de₂ and T(f₃)=ge₁+he₂, where a, b, c, d, g, h∈R. Then for x∈V, there exist r, s, t∈R such that x=rf₁+sf₂+tf₃. Thus define T(x)= rT(f₁)+sT(f₂)+tT(f₃), because T has to be linear.

3.1. Example:

Consider A=

$$\begin{pmatrix}1 & -1 & 0 \\\ 0 & 1 & 1\end{pmatrix}$$

Then find the associated linear transformation of A w.r.t. the basis B₁={(1, 0, 0), (-1, 1, 0), (0, 1, 1)} and B₂={(-1, 1), (0, 1)} of R³ and R² respectively.
Define T(1, 0, 0)=1(-1, 1)+0(0, 1)=(-1, 1), T(-1, 1, 0)=-1(-1, 1)+1(0, 1)=(1, 0) and T(0, 1, 1)=0(-1, 1)+1(0, 1)=(0, 1). Since (x, y, z)=a(1, 0, 0)+b(-1, 1, 0)+c(0, 1, 1), hence a-b=x, b+c=y and c=z. By solving these equations we get, a=x+y-z, b=y-z and c=z.
Now, define T:R³→R² by T(x, y, z)= aT(1, 0, 0)+bT(-1, 1, 0)+cT(0, 1, 1)=a(-1, 1)+b(1, 0)+c(0, 1)=(-a+b, a+c)=(-x, x+y), where x, y∈R.
The linear transformation T:R³→R² associated with the matrix 2×3 A w.r.t. the basis B₁ and B₂ is T(x, y, z)=( -x, x+y), where x, y∈R.