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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:

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!