Define linear transformation with an example.Let A = $$\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}$$, v = $$\begin{bmatrix}2\\ -1\end{bmatrix}$$, b = $$\begin{bmatrix}3\\ 2\\ 4\end{bmatrix}$$, x = $$\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$and define a transformation T:R2 → R2 by T(x) = Ax then find T(v) find x ∈ R2 whose image under T is b

Let T:V → W be a tranformation (mapping or function) such that,

1. T(c, v) = c . T(v)
2. T(u + v) = T(u) + T(v)  ∀ c ∈ K and u, v ∈ V

Example: Let A = (aij)m × n be an m × n matrix.

Let T:Rn → Rm

T: multiplication be the matrix A

i.e. T(x) = Ax is a linear transformation

Solution:

Given Matrices are

A = $$\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}$$

v = $$\begin{bmatrix}2\\ -1\end{bmatrix}$$

b = $$\begin{bmatrix}3\\ 2\\ 4\end{bmatrix}$$

x = $$\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$

Given  that transformation T:R2 → R2 by T(x) = Ax

a) First Part:

T(u) = Au

= $$\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}$$ $$\begin{bmatrix}2\\ -1\end{bmatrix}$$

= $$\begin{bmatrix}2+3\\ 6-5 \\-2-7\end{bmatrix}$$

= $$\begin{bmatrix}5\\ 1 \\-9\end{bmatrix}$$

b) Second Part:

Let x = $$\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$

Suppose x in R2 where image under T is b then

T(x) = b

Ax = b

$$\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}$$$$\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$ = $$\begin{bmatrix}3\\ 2\\ 4\end{bmatrix}$$

The augmented matrix of Ax = b is

$$\begin{bmatrix}1 & -3 & : & 3\\ 3 & 5 & : & 2\\ -1 & 7 & : & 4\end{bmatrix}$$

R2 → R2 – 3R1,   R3 → R3 + R1

$$\begin{bmatrix}1 & -3 & : & 3\\ 0 & 14 & : & -7\\ 0 & 4 & : & 7\end{bmatrix}$$

R2 → R2 / 14

$$\begin{bmatrix}1 & -3 & : & 3\\ 0 & 1 & : & \frac{-1}{2}\\ 0 & 4 & : & 7\end{bmatrix}$$

R3 → R3 – 4R2,   R1 → R1 + 3R2

$$\begin{bmatrix}1 & 0 & : & \frac{3}{2}\\ 0 & 1 & : & \frac{-1}{2}\\ 0 & 0 & : & 9\end{bmatrix}$$

This implies $$x_1 = \frac{3}{2}$$ and $$x_2 = \frac{-1}{2}$$

Thus, x = $$\begin{bmatrix}\frac{3}{2}\\ \frac{-1}{2}\end{bmatrix}$$ in R2 whose image under T is b.