Linear Transformations and Geometry

Linear Transformations and Geometry

Linear algebra is not just about numbers in tables—it’s about transformations of space. Matrices can be seen as functions that transform vectors, stretching, rotating, reflecting, or projecting them. Understanding these transformations geometrically is essential for intuition in machine learning.

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Linear Transformations

A function $T: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation if it satisfies:

1. Additivity: $T(x + y) = T(x) + T(y)$
2. Homogeneity: $T(cx) = cT(x)$ for scalar $c$

Every linear transformation can be represented as a matrix multiplication:

$$T(x) = A x$$

for some matrix $A$.

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Geometric Interpretations

1. Scaling – Multiply vectors by a scalar.
   Example: $\begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}$ scales $x$ by 2 and $y$ by 3.

2. Rotation – Preserve length but rotate direction.
   Example (2D rotation by $\theta$):

   $$R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$

3. Reflection – Flip across a line or plane.

4. Projection – Collapse vectors onto a subspace.
   Example: Project onto $x$-axis in 2D:

   $$P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$

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Mini Examples

• Stretching:

  $$A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$$

  doubles the $x$-axis component while leaving $y$ unchanged.

• Rotation by 90°:

  $$R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$

  rotates any vector counterclockwise by 90°.

• Projection:

  $$P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$

  projects vectors onto the $x$-axis.

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Hands-on with Python and Rust

::: code-group

import numpy as np

v = np.array([1, 2])

# Scaling
A = np.array([[2, 0], [0, 3]])
scaled = A.dot(v)

# Rotation (90 degrees)
R = np.array([[0, -1], [1, 0]])
rotated = R.dot(v)

# Projection onto x-axis
P = np.array([[1, 0], [0, 0]])
projected = P.dot(v)

print("Original vector:", v)
print("Scaled:", scaled)
print("Rotated:", rotated)
print("Projected:", projected)

use ndarray::{array, Array1, Array2};

fn main() {
    let v: Array1<f64> = array![1.0, 2.0];

    // Scaling
    let a: Array2<f64> = array![[2.0, 0.0], [0.0, 3.0]];
    let scaled = a.dot(&v);

    // Rotation (90 degrees)
    let r: Array2<f64> = array![[0.0, -1.0], [1.0, 0.0]];
    let rotated = r.dot(&v);

    // Projection onto x-axis
    let p: Array2<f64> = array![[1.0, 0.0], [0.0, 0.0]];
    let projected = p.dot(&v);

    println!("Original vector: {:?}", v);
    println!("Scaled: {:?}", scaled);
    println!("Rotated: {:?}", rotated);
    println!("Projected: {:?}", projected);
}

:::

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Connection to ML

• Scaling → standardization of features.
• Rotation → PCA rotates data into new basis.
• Projection → dimensionality reduction (projecting onto lower-dimensional subspaces).
• Reflections & symmetries → used in data augmentation and transformations.

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Next Steps

Continue to Subspaces and Basis.