Linear Transformations & Matrices

Essence of Linear Algebra · Chapter 3 🔗 https://youtu.be/kYB8IZa5AuE

🧠 Big Idea

A matrix is not just a grid of numbers — it's a description of how space gets moved and reshaped. Every 2×2 matrix encodes a linear transformation, and the two columns tell you exactly where the basis vectors î and ĵ land after the transformation.

🔄 What Is a Linear Transformation?

A transformation is a function: takes in a vector, spits out another vector. We use the word "transformation" instead of "function" because it suggests movement — every input vector moves to its output position.

Two rules that make it "linear"

Lines remain lines — no curves, no bending
Origin stays fixed — the zero vector maps to itself

Equivalent visual rule: grid lines stay parallel and evenly spaced after the transformation (though angles between them may change).

Non-linear examples to recognize:

Lines get curved → not linear
The origin moves → not linear

📐 Why Tracking Just î and ĵ Is Enough

Since every vector v = x·î + y·ĵ, and linear transformations preserve addition and scaling:

L(v) = L(x·î + y·ĵ)
     = x·L(î) + y·L(ĵ)

So once you know where î and ĵ land, you know where every vector lands. A 2D linear transformation is completely pinned down by just 4 numbers.

🔢 Matrices = Packaged Transformation Info

Pack the landing spots of î and ĵ into columns:

Matrix = [ L(î) | L(ĵ) ] = | a  b |
                             | c  d |

Column 1 = where î lands = [a, c]
Column 2 = where ĵ lands = [b, d]

Matrix-vector multiplication (the geometric meaning)

| a  b | · | x |  =  x · | a |  +  y · | b |  =  | ax + by |
| c  d |   | y |          | c |          | d |     | cx + dy |

Not a formula to memorize — it's just "take x copies of the new î, plus y copies of the new ĵ."

🎯 Key Transformation Examples

Transformation	Matrix	What happens
90° counterclockwise rotation	`[[0, -1], [1, 0]]`	î → [0,1], ĵ → [-1,0]
Shear	`[[1, 1], [0, 1]]`	î stays, ĵ → [1,1]
Horizontal flip	`[[-1, 0], [0, 1]]`	î → [-1,0], ĵ stays
Scale by 2	`[[2, 0], [0, 2]]`	everything doubles
Squish to a line	`[[1, 2], [2, 4]]`	columns dependent → 2D → 1D

⚠️ Linearly Dependent Columns

If the two columns of a matrix are linearly dependent (one is a scalar multiple of the other), the transformation squishes all of 2D space onto a single line. The entire plane collapses to 1D.

💡 Key Takeaways

Matrices are transformations — not just number grids
Linear = lines stay lines + origin stays fixed = grid stays parallel & evenly spaced
Columns of the matrix = where the basis vectors land
Matrix × vector = "where does this vector end up?"
4 numbers are all you need to fully describe any 2D linear transformation
linear transformations are a way to move around space such that:
- gird lines remain parallel and evenly spaced
- origin remains fixed

🔗 Series

← Ch.2 – Span & Basis → Ch.4 – Matrix multiplication as composition