📐 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 × 5 = 15

Scaling in 2D (Vector Form)

Vector Representation

v⃗ = [ x
       y ]

Scaling Formula


T(v⃗) =
[ k   0
  0   k ]
[ x
  y ]
=
[ kx
  ky ]

Example (Step-wise)

Given:

v⃗ =
[ 2
  3 ],   k = 2

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

Resultant vector:

[ 4
  6 ]

Scaling with Different Factors (Non-uniform Scaling)

Sometimes x and y scale differently.

Formula

T(v⃗) =
[ kₓ   0
  0    kᵧ ]
[ x
  y ]
=
[ kₓx
  kᵧy ]

Example

Given:

v⃗ =
[ 2
  4 ],   kₓ = 3,   kᵧ = 1

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

Result:

[ 6
  4 ]

Scaling in 3D

Matrix Form

[ kₓ   0    0
  0    kᵧ   0
  0    0    k_z ]
[ x
  y
  z ]

Example

v⃗ =
[ 1
  2
  3 ],   (kₓ, kᵧ, k_z) = (2, 1, 3)

Result:

[ 2
  2
  9 ]

← Previous Next →