System of Linear Equations

What is a System of Linear Equations?

A collection of linear equations with the same set of variables, typically written as Ax = y, where A is a matrix of coefficients, x is the vector of unknowns, and y is the result vector.

What is Gaussian Elimination?

A method for solving systems of linear equations that involves transforming the system’s augmented matrix into row echelon form (REF) using elementary row operations, then performing back-substitution to find solutions.

What is a Homogeneous Linear System?

A system of equations where the right-hand side is the zero vector: Ax = 0. Always has at least one solution (the zero vector).

How can we express the General Solution to a linear system Ax = y?

General Solution = Particular solution x_0 + General Solution to Ax= 0. That is: x = x_0 + \sum^{k}_{i=0}\alpha_i v_i where \{v_i\} is a basis for the null space (kernel) of A, and \alpha_i\in F.

What is the Geometric Interpretation of the solution set to a system Ax=y?

It is either:

• Empty (inconsistent system),

• A point (unique solution),

• A line, plane or hyper plane (infinite solutions forming an affine subspace)

What is the Rank-Nullity Theorem?

For any matrix A\in\mathbb{F}^{-xn}, dim(Ker(A)) + dim(Im(A)) = n. That is, nullity + rank = number of variables.

When does a linear system Ax=y have a Solution?

When y\in Im(A), i.e. the vector y can be expressed as a linear combination of the columns of A.

What is the Matrix Form of a linear system?

It can be written compactly as Ax = y where:

• A\in\mathbb{F}^{pxn}

• x\in\mathbb{F}^n

• y\in\mathbb{F}^p

What are Free Variables in a linear system?

Variables not leading any row in the row echelon form. They can take arbitrary values, resulting in infinite solutions.

What is a Particular Solution?

A single solution x_0 that satisfies Ax=y. Used to express the general solution as x = x_0 + Ker(A).