,

Contents · Linear algebra (vectors, matrices, rank, determinants)

Overview

Summary

Linear algebra studies vector spaces and linear maps. Core notions include span/basis, dimension, rank, determinants, and solving linear systems via Gaussian elimination.

Details

Concepts
  • Vectors and matrices; linear combinations, span, independence, basis, dimension.
  • Matrix operations: addition, multiplication, transpose; linear maps and matrix representation.
  • Rank and nullity; rank–nullity theorem; solving Ax = b.
  • Determinants: properties, geometric meaning (volume scaling), cofactor expansion.
  • Invertibility: A is invertible iff det(A) ≠ 0 iff rank(A) = n.
  • Row reduction, echelon forms, and Gaussian elimination.

Exercises

Hands-on
  1. Find a basis and dimension of the span of given vectors in R^3.
  2. Compute rank(A) and a basis for Null(A) for a 3×4 matrix.
  3. Use determinants to test invertibility and compute A^{-1} when it exists.

©