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✅ Vectors describe positions or directions within a coordinate space.
✅ Matrices describe how to transform vectors from one space to another.

[ Xaxis.x  Yaxis.x  Zaxis.x  Tx ]
[ Xaxis.y  Yaxis.y  Zaxis.y  Ty ]
[ Xaxis.z  Yaxis.z  Zaxis.z  Tz ]
[   0        0        0      1 ]

A 4×4 transformation matrix is made up of:

• 3 direction vectors (X, Y, Z axes — orientation)

• 1 position vector (T — translation)

• 1 extra row that supports homogeneous coordinates

YES — asking what is a basis means you’re ready to understand all of 3D spacebuilding and matrix logic. Let’s break it down clearly, intuitively, and practically for tech art 👇

What Is a Basis?

A basis is a set of vectors that define a coordinate space.
It tells you:

• Where the origin is

• What direction X, Y, Z go

• What "1 unit" means in each direction

It’s like building a mini coordinate system — your own local grid.

✅ Example: Standard Basis in 3D

The default basis in world space is:

X = (1, 0, 0)
Y = (0, 1, 0)
Z = (0, 0, 1)

This is called an orthonormal basis:

• Each axis is unit length

• Each axis is perpendicular to the others

This is what Unity uses for world space, object space (by default), and more.

🧩 Why Is a Basis Useful?

A basis lets you build your own space inside another space.

You can:

• Rotate it (to align to a surface)

• Scale it (to stretch/compress space)

• Offset it (with a position = origin)

• Pack it into a matrix → for transforming vectors

🧱 Basis = The Building Blocks of a Matrix

In a transformation matrix, the 3 main columns are your basis vectors:

[ Right  Up  Forward  Position ]
[   X     Y     Z        T     ]

So:

• The first 3 columns = your basis vectors (local X, Y, Z)

• The 4th column = your new origin (translation vector)

✅ A matrix is just a basis with a location.

🧪 Real Tech Art Examples

✅ A basis is a set of vectors that define a space's axes.
A matrix = basis + position.
You use it to move, rotate, or remap things between spaces.

• Identity Matrix

  • Does nothing when applied to a vector.

  • Basis for understanding other transformations.

  • Represents standard world coordinates.

• Uniform Scaled Matrix

  • Modify all diagonal values equally.

  • Scales objects equally in all directions.

  • E.g., 0.5 on all diagonal entries = mesh is scaled down by half.

• Axis Scaled Matrix

  • Change individual diagonal values independently.

  • Scales the object along specific axes.

  • E.g., a 3 in the X-axis = stretches object 3x in X only.

• Mirror (Reflection) Matrix

  • Multiply a diagonal entry by -1 to mirror on that axis.

  • E.g., -1 in X = flips mesh on the X-axis.

• Matrix with Normalized Basis

  • Basis vectors (matrix columns) are unit length and orthogonal.

  • Often used for pure rotation without scaling.

  • Each axis remains perpendicular and consistent in size.

• Matrix with Non-Orthogonal Basis

  • Columns are not perpendicular.

  • Causes skewing—object axes are no longer at right angles.

• Matrix with Non-Normalized Basis

  • Basis vectors may differ in length.

  • Can scale along arbitrary axes (not just X, Y, Z).

• Rotation Matrices

  • Represented by changing off-diagonal values (not just diagonal).

  • Rotate vectors in space while maintaining their magnitude.

• Matrix Interpretation Tip

  • Think of each matrix column as defining a new space axis.

  • Multiplying a vector by a matrix expresses it in that new space.

Let me know if you’d like a diagram or Unity version of any of these!