When Algebra Becomes Translation

At first, the Change of Basis Matrix looked like pure exam gymnastics — another symbolic trick professors used to stretch our patience. But one quiet night during my Master’s, something clicked.

Changing bases wasn’t about manipulating data; it was about translating meaning. It was the mathematical version of switching camera angles — same scene, new clarity.

Perspective, I realized, doesn’t come from adding data; it comes from rotating frames.

“The change of basis doesn’t alter truth — it changes how truth is seen.”

That revelation would go on to shape not just my understanding of AI embeddings, but how I navigated every career pivot since.

What Changing a Basis Really Means

Every vector space lets you describe points in multiple coordinate systems.
A basis is simply the set of reference vectors you use for description.

Suppose you have two bases:

The change of basis matrix P converts coordinates from one frame to the other.
Each column of P represents a vector of the new basis C written in terms of the old basis BBB.

Mathematically:

and therefore,

When a transformation A is expressed in basis B, its representation in basis C becomes:

This relationship, humble as it seems, powers diagonalization, PCA, and modern neural embeddings — every place where clarity is born from rotation.

Python Demo – Changing Coordinates

import numpy as np

# Basis B (standard)
B = np.eye(2)

# New basis C
C = np.array([[1, 1],
              [1, -1]])

# Change of basis matrix from C to B
P = C
P_inv = np.linalg.inv(P)

# Vector in basis B
v_B = np.array([[2],
                [1]])

# Convert to basis C
v_C = P_inv @ v_B

# Convert back to basis B
v_recovered = P @ v_C

print("Coordinates in basis C:\n", v_C)
print("Recovered vector in basis B:\n", v_recovered)

The result shows the same vector — just described differently.
Mathematics calls it equivalence; we call it perspective.

When It Finally Clicked

I remember pacing around my apartment, caffeine fading, equations sprawling across three whiteboards.
The columns of the matrix stopped being numbers — they became directions of meaning.
Each column wasn’t data; it was a new lens.

That’s when I realized: the change of basis matrix is math’s translator.
It doesn’t change what exists — it changes how it’s understood.

“Mathematics isn’t hiding wisdom; it encodes it in symmetry.”

That night, I understood not only the proof —
but why transformation and translation are the same idea expressed in different coordinates.

From PCA to Word Embeddings — Where AI Changes Its Basis

Once you start looking for it, the change of basis hides everywhere in AI.

Principal Component Analysis (PCA)

PCA rotates data into a new basis where axes (principal components) align with directions of maximum variance.
The covariance matrix

is diagonalized via eigen decomposition:

where D contains eigenvalues (variance along each axis).
Transforming data via X' = XP expresses it in the new, uncorrelated basis.

from sklearn.decomposition import PCA
import numpy as np

X = np.random.randn(200, 3)
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

print("PCA Components:\n", pca.components_)

Those pca.components_ are nothing but the columns of P — the new basis vectors.

Word Embeddings

Language models do the same — they learn coordinate systems for meaning.
Words that used to exist as discrete symbols get represented as continuous vectors in a high-dimensional space.
Changing context (via fine-tuning or new corpora) effectively applies a new basis, reorienting meaning while preserving relationships.

That’s why analogies like

work — they remain invariant under the right basis.

Startup Geometry — Reframing Failure

Every failed startup I’ve been part of was, in hindsight, a coordinate misalignment.
We weren’t wrong — we were projecting in the wrong space.

When OXOFIT pivoted from “tech-first” to “community-first,” we didn’t change the product — we changed the basis of value.
Metrics that looked meaningless before suddenly made sense once rotated toward empathy and retention instead of pure acquisition.

You don’t always need new data; sometimes you need new axes.

“You can’t fix what you can’t measure correctly — so redefine the coordinates.”

Deriving the Change of Basis Step by Step

It’s worth grounding the intuition in derivation — because every line of algebra reveals something deeper.

Let

Each Ci​ can be expressed as a combination of bj:

The coefficients aji​ form the columns of matrix P, meaning:

C = BP

For a vector v:

If A is a transformation matrix under B:

This operation is the algebraic foundation of diagonalization — the art of simplifying complexity through rotation.

Python Demo – Diagonalization Through Basis Change

import numpy as np

A = np.array([[4, 1],
              [2, 3]])

# Eigen decomposition
eigvals, eigvecs = np.linalg.eig(A)

# Change of basis
P = eigvecs
P_inv = np.linalg.inv(P)
A_prime = P_inv @ A @ P

print("Diagonal form:\n", A_prime)

The result?
A diagonal matrix where dependencies disappear — the purest form of simplicity hidden inside complexity.

Coding Intuition Into Muscle Memory

Equations alone don’t build intuition — interaction does.
I used to write small scripts that rotated 2×2 transformation matrices repeatedly, plotting results live.
Watching coordinate frames twist while the vector stayed the same built geometric intuition faster than a dozen proofs.

Eventually, I could feel what

meant before computing it.
That’s the difference between knowing math and thinking in math.

Rebuilding After Layoffs — Life’s Change of Basis

When layoffs hit in my career, I thought everything collapsed.
In truth, nothing was lost — the basis had changed.

The same skills that once belonged to legacy systems (COBOL, mainframes) re-expressed themselves through modern coordinates (cloud, AI).
Magnitude unchanged — direction updated.

Each reinvention since has been a new coordinate projection.
And mentoring engineers now, I teach them this:
Don’t erase your past — transform your basis.
What looks like failure might just be data described in the wrong frame.

“Growth isn’t starting over — it’s re-expressing yourself in the right coordinate system.”

Quiet Recap

• The change of basis converts representation — not reality.

• A' = P^{-1}AP expresses the same transformation from a new lens.

• PCA and embeddings use it to extract clarity from chaos.

• Rotating the basis simplifies complexity — in math, AI, and self.

• Every reinvention is just a reprojection of old vectors onto new axes.

Because at the core, every transformation worth making — in models or in life — is a change of basis done right.

~ BitByBharat
Where math meets meaning, and equations learn empathy.