📐 Linear Transformation - Scaling

Last Updated: Jan 2026

Scaling is the most basic linear transformation, where a vector is stretched or shrunk. Scaling changes the magnitude (length) of a vector, not its direction.

🗣 Hinglish Tip: Scaling = vector ko bada ya chhota karna, direction same rehta hai

Scaling in 1D

Formula

For a scalar k and vector x:

 $T(x) = kx$

Example

Given:

 $x = 5, k = 3$

 $T(x) = 3 \times 5 = 15$

Scaling in 2D (Vector Form)

Vector Representation

 $\vec{v} = \begin{bmatrix} x \\ y \end{bmatrix}$

Scaling Formula

 $T(\vec{v}) = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} kx \\ ky \end{bmatrix}$

Example (Step-wise)

Given:

 $\vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \qquad k = 2$

Component	Calculation	Result
x	2 x 2	4
y	3 x 2	6

Resultant vector:

 $\vec{v}' = \begin{bmatrix} 4 \\ 6 \end{bmatrix}$

Scaling with Different Factors (Non-uniform Scaling)

Sometimes x and y scale differently.

Formula

 $T(\vec{v}) = \begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} k_xx \\ k_yy \end{bmatrix}$

Example

Given:

 $\vec{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}, \qquad k_x = 3, \qquad k_y = 1$

Axis	Calculation	Result
x-axis	3 x 2	6
y-axis	1 x 4	4

Result:

 $\begin{bmatrix} 6 \\ 4 \end{bmatrix}$

Scaling in 3D

Matrix Form

 $\begin{bmatrix} k_x & 0 & 0 \\ 0 & k_y & 0 \\ 0 & 0 & k_z \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

Example

 $\vec{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \qquad (k_x,\,k_y,\,k_z) = (2,\,1,\,3)$

Result:

 $\begin{bmatrix} 2\\ 2\\ 9 \end{bmatrix}$

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