📐 What is Linear Algebra?
Linear algebra is the branch of mathematics dealing with vectors, matrices, and linear transformations. It's essential for understanding systems of equations, transformations in space, and many algorithms in computer science and AI.
🧱 Core Concepts:
- Scalars – Single numbers (e.g., 3, -2.5)
- Vectors – Ordered lists of numbers (e.g., 2,4,62, 4, 6)
- Matrices – 2D arrays of numbers, like a grid (e.g., \[1,2\[1, 2, 3,43, 4])
- Tensors – Multidimensional generalizations of matrices (used in deep learning)
🔄 Key Operations:
- Addition/Subtraction of vectors or matrices
- Matrix multiplication (used in transformations and solving equations)
- Dot product and cross product
- Transpose (flipping rows and columns)
- Inverse (used to "undo" matrix operations)
- Determinant (used to check matrix properties)
📊 Applications:
- Solving systems of equations (e.g., linear regression)
- Graphics & transformations (e.g., 3D modeling, rotations)
- Machine Learning (e.g., neural networks use matrix ops for weights & activations)
- Data compression (e.g., PCA uses eigenvectors and eigenvalues)
💡 Real Example (Simple):
Solve the system of equations:
2x + 3y = 8 x - y = 2
This can be written in matrix form:
AX = B
[231−1]⋅[xy]=[82]\begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 2 \end{bmatrix}
Solve using matrix inversion or row reduction.
Want to see a visual example or a Python implementation using NumPy?