Linear Algebra Cheat Sheet

Key Definitions

• Vector: An ordered list of numbers representing a point in space. ✅

• Matrix: A rectangular array of numbers arranged in rows and columns. ✅

• Determinant: A scalar value that can be computed from the elements of a square matrix. ✅

• Eigenvalue: A scalar that indicates how much a corresponding eigenvector is stretched during a linear transformation. ✅

• Linear Transformation: A mapping between two vector spaces that preserves vector addition and scalar multiplication. ✅

Important Concepts

Vector Operations

• Addition: Combine corresponding components.

• Scalar Multiplication: Multiply each component by a scalar.

Matrix Operations

• Matrix Addition: Add corresponding elements.

• Matrix Multiplication: Dot product of rows and columns.

Systems of Equations

• Can be represented in matrix form as AX = B, where:

• A = coefficient matrix

• X = variable matrix

• B = constant matrix

Key Formulas

• Matrix Multiplication:

C = AB where C[i][j] = Σ A[i][k] * B[k][j] for all k. ✅

• Determinant of 2x2 Matrix:

det(A) = ad - bc for matrix A = [[a, b], [c, d]]. ✅

• Eigenvalue Equation:

Av = λv where A is a matrix, v is the eigenvector, and λ is the eigenvalue. ✅

Common Mistakes to Avoid

Common Mistake: Confusing matrix multiplication with element-wise multiplication. Remember that matrix multiplication is not commutative.

Memory Tricks

Tip: "VIM" — Vectors, Inverses, Matrices. Remember these key concepts for quick recall.

Key Diagrams

Matrix A          Vector x              Result Ax
| a11 a12 |    *  | x1 |    =  | r1 |
| a21 a22 |       | x2 |       | r2 |

Eigenvalue Representation:
A  v = λ  v