The present experiment shows that every linear transformation can be written using a matrix and vice-versa. This makes these notions useful to work with. The experiment helps students understand what a linear transformation is and how it is related to matrices. Here, students also learn to apply these concepts in various contexts in a simple and clear way.

1. Linear transformation:

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

1.2. Zero transformation:

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

1.3. Example:

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). This completes this proof.

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, 0) and T(y₁, y₂)=(y₁, 0). 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). This completes this proof.

c). Let T:R² → R² such that T(x, y)=(y, x), where x, y∈R. Then T is linear.
Proof: Let x=(x₁, x₂) and y=(y₁, y₂) ∈R². Then T(x₁, x₂)=(x₂, x₁) and T(y₁, y₂)=(y₂, y₁). 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)=(αx₂, αx₁) and αT(x)=α(x₂, x₁)=(αx₂, αx₁). This completes this proof.

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 F≡R or C 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, …, r₂ ∈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 bases 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. Set A= $\begin{pmatrix}a & c & e \\\ b & d & f\end{pmatrix}$. Then matrix A is of order 2×3 and is called the matrix represented by T w.r.t. the bases B₁ and B₂. In a similar way, one can check that the matrix represented by 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 we find the matrix associated with the transformation w.r.t. the bases B₁={(1, 0), (0,1)} and B₂={(0, -1), (-1, 0)}.
Clearly 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 represented by T w.r.t. the bases B₁ and B₂ is A = $\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$.

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 bases B₁={f₁, f₂, f₃} and B₂={e₁, e₂} of R³ and R² respectively 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. 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 we find the associated linear transformation of A w.r.t. the bases 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 bases B₁ and B₂ is given by T(x, y, z)=(-x, x+y), where x, y∈R.