A linear transformation is a function T from ℝⁿ to ℝᵐ that keeps vector addition and scalar multiplication: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalars c. To check one, test both properties with general vectors, or find one counterexample to show it is not linear.
Linear independence means no vector in a set can be built from the others. Vectors v1, …, vp are linearly independent when the only solution of c1v1 + … + cpvp = 0 is c1 = … = cp = 0. To test, put the vectors in the columns of a matrix and row reduce: they are independent exactly when every column has a pivot.
Parametric vector form writes every solution of a linear system as x = p + t₁v₁ + ⋯ + tₖvₖ, with one vector for each free variable. To find it, reduce the system, solve for the pivot variables, and split x by free variable. For Ax = b, p is one particular solution; for Ax = 0, p = 0.
A homogeneous system is a linear system Ax = 0, where every constant on the right side is 0. It always has the trivial solution x = 0. It has a nontrivial solution, one with some entry not 0, exactly when the system has at least one free variable. That always happens when there are more unknowns than equations.
The matrix equation Ax = b writes a whole linear system with one matrix A and one vector of unknowns x. The product Ax is the linear combination of the columns of A that uses the entries of x as weights. So Ax = b has a solution exactly when b is in the span of the columns of A.
The span of vectors v₁, v₂, …, vₚ is the set of all their linear combinations c₁v₁ + c₂v₂ + ⋯ + cₚvₚ. A vector b is in the span exactly when the augmented matrix [v₁ ⋯ vₚ | b] is consistent. The vectors span all of ℝⁿ when their matrix has a pivot in every row.
A linear combination of vectors v₁, v₂, …, vₚ is any sum c₁v₁ + c₂v₂ + ⋯ + cₚvₚ, with scalar weights c₁, …, cₚ. To find whether a vector b is a linear combination of given vectors, solve the vector equation x₁v₁ + ⋯ + xₚvₚ = b by row reducing the augmented matrix [v₁ ⋯ vₚ | b].
Applications of linear systems turn real problems into equations and solve them by row reduction. To balance a chemical equation, write one equation per element so atoms are conserved. To analyze network flow, write one equation per junction so flow in equals flow out. Free variables then describe every possible answer.
Reduced row echelon form (RREF) is a row echelon form in which every pivot is 1 and is the only nonzero entry in its column. Gauss-Jordan elimination reaches it by also clearing the entries above each pivot, so the solution of a linear system can be read straight from the matrix. Every matrix has exactly one RREF.
Gaussian elimination solves a system of linear equations by using row operations to reduce its augmented matrix to row echelon form. The pivot positions in that form show whether the system has no solution, exactly one solution or infinitely many, and back substitution then gives the solutions.