Linear_Algebra // Mod_08

ORTHOGONALITY

The mathematics of perfect independence. Discovering perpendicularity in N-dimensional space through projections and the Gram-Schmidt process.

01 // The 90° Rule

In basic geometry, two lines are perpendicular if they meet at 90°. In linear algebra, we call vectors Orthogonal. Because they share absolutely zero directional overlap, they are perfectly independent of one another.

The ultimate test for orthogonality is the Dot Product. If the dot product of two vectors is exactly zero, they are completely orthogonal.

u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0

02 // The Shadow (Projection)

If vectors aren't orthogonal, you can force them to be! A Projection is like shining a flashlight straight down onto a vector to find its "shadow" along another line. The math calculates the exact scalar needed to stretch the base vector so that the error line drops perfectly at a 90° angle.

Projection Engine

Target Vector (v)[3, 4]
Base Vector (u)[5, 1]
Scalar Multiplier
v⃗⋅u⃗u⃗⋅u⃗=1926≈0.73\frac{\vec{v} \cdot \vec{u}}{\vec{u} \cdot \vec{u}} = \frac{19}{26} \approx 0.73
Target v⃗\vec{v}
Base u⃗\vec{u}
Projection

03 // The Purifier (Gram-Schmidt)

The Gram-Schmidt Process

Standard bases in the real world are often messy and skewed. The Gram-Schmidt process is an algorithm that takes any valid basis and systematically "purifies" it into an Orthonormal Basis (where every vector is 90° to all others, and scaled to a length of exactly 1).

Step 1: The Anchor

Take the first vector exactly as it is. It becomes the anchor for the entire new coordinate system.

u⃗1=v⃗1\vec{u}_1 = \vec{v}_1
Step 2: Subtraction

Take the second vector, project it onto the anchor, and subtract the shadow. You are left with only the purely perpendicular component.

u⃗2=v⃗2−proju⃗1(v⃗2)\vec{u}_2 = \vec{v}_2 - \text{proj}_{\vec{u}_1}(\vec{v}_2)
Step 3: Normalize

Once all vectors are perfectly orthogonal, divide each one by its own magnitude to shrink them down to unit length (1).

e^i=u⃗i∣∣u⃗i∣∣\hat{e}_i = \frac{\vec{u}_i}{||\vec{u}_i||}
Result: A perfect Orthonormal Basis

Independence Proven

You are ready to compress matrices using SVD.

Next: SVD