Linear_Algebra // Mod_05

TRANSFORMS

Matrices are functions. Discover how multiplying a vector by a matrix physically warps, rotates, and shears the fabric of space itself.

01 // The Machine

Every matrix acts as a mapping engine. If you feed it a coordinate x⃗\vec{x}, it outputs a brand new coordinate. We define this mapping as T(x⃗)=Ax⃗T(\vec{x}) = A\vec{x}.

For a transformation to be strictly Linear, it must follow two physical rules:

1
All grid lines must remain straight and parallel. No curving.
2
The Origin (0,0) must stay permanently locked in place.

02 // The Execution

The columns of a transformation matrix tell you exactly where the fundamental basis vectors i^\hat{i} (x-axis) and j^\hat{j} (y-axis) will land. The entire rest of the grid simply follows them!

Spatial Warper

Transformation Matrix
New i^\hat{i} location
New j^\hat{j} location

03 // The Standard Dictionary

Resize
Scaling

Modifying the main diagonal stretches or shrinks the space. If both values are the same, it scales uniformly.

[sx00sy]\begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}
Slant
Shearing

Pushing the non-diagonal elements tilts the axes. It turns squares into slanted parallelograms without changing their area.

[1k01]\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}
Spin
Rotation

The ultimate trigonometric preset. It rotates the entire grid by an exact angle θ\theta counter-clockwise.

[cos⁡θ−sin⁡θsin⁡θcos⁡θ]\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Spatial Control Acquired

You are ready to command n-dimensional Vector Spaces.

Next: Vector Spaces