" [...] a collection of one or more linear equations involving the same variables."
A system of linear equations is a collection of linear equations that are meant to be solved simultaneously. The solution to a system of linear equations can be represented as a column matrix.
Systems of Linear Equations: A set of linear equations in which each equation has the same variables with their corresponding coefficients.
Matrix Representation: A way of representing systems of linear equations using matrices and vectors.
Row Echelon Form: A way of solving systems of linear equations by transforming the system into a simplified form with a unique solution.
Gaussian Elimination: A method of solving systems of linear equations by row reduction of an augmented matrix.
Determinants: A scalar quantity associated with a square matrix, used to determine the invertibility of a matrix.
Cramer's Rule: A method for solving systems of linear equations using determinants.
Linear Independence: A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others.
Span: The set of all linear combinations of a set of vectors.
Basis: A set of linearly independent vectors that span a subspace.
Rank: The rank of a matrix is the number of linearly independent rows or columns of the matrix.
Null Space: The set of all solutions to the homogeneous equation Ax = 0.
Inverse Matrices: A matrix that, when multiplied by another matrix, gives the identity matrix.
Linear Transformations: A function that maps vectors from one vector space to another, preserving their linearity.
Eigenvalues and Eigenvectors: A scalar and its corresponding vector that satisfy the equation Ax = λx, where A is a square matrix.
Eigenspaces: The set of all eigenvectors corresponding to a single eigenvalue.
Consistent: A system of linear equations is consistent if it has at least one solution.
Inconsistent: A system of linear equations is inconsistent if it has no solution.
Dependent: A system of linear equations is dependent if it has infinitely many solutions.
Independent: A system of linear equations is independent if it has a unique solution.
Homogeneous: A system of linear equations is homogeneous if all its constant terms are equal to zero.
Non-homogeneous: A system of linear equations is non-homogeneous if one or more of its constant terms are non-zero.
Gaussian: A system of linear equations is in Gaussian form if it has been reduced to upper triangular form (or row echelon form) using Gauss-Jordan elimination.
Augmented: A system of linear equations in augmented form includes the coefficients and constant terms of the equations on the left-hand side of a vertical bar and the solutions on the right-hand side of the bar.
Cramer's rule: A method of solving a system of linear equations using determinants.
Square: A system of linear equations is square if the number of equations equals the number of unknowns.
Underdetermined: A system of linear equations is underdetermined if it has fewer equations than unknowns.
Overdetermined: A system of linear equations is overdetermined if it has more equations than unknowns.
Trivial: A solution is trivial if all unknowns are equal to zero.
Non-trivial: A solution is non-trivial if at least one unknown is non-zero.
Singular: A system of linear equations is singular if its coefficient matrix is non-invertible.
Non-singular: A system of linear equations is non-singular if its coefficient matrix is invertible.
Rank deficient: A system of linear equations is rank deficient if its rank is less than the number of unknowns.
Rank sufficient: A system of linear equations is rank sufficient if its rank equals the number of unknowns.
"A system of three equations in the three variables x, y, z."
"A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied."
"(x,y,z)=(1,-2,-2), since it makes all three equations valid."
"Linear systems are the basis and a fundamental part of linear algebra, a subject used in most modern mathematics."
"Computational algorithms for finding the solutions are an important part of numerical linear algebra."
"They play a prominent role in engineering, physics, chemistry, computer science, and economics."
"A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system."
"The coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers."
"The theory and the algorithms apply for coefficients and solutions in any field."
"For solutions in an integral domain like the ring of integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring."
"Integer linear programming is a collection of methods for finding the 'best' integer solution (when there are many)."
"Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials."
"Also tropical geometry is an example of linear algebra in a more exotic structure." Note: The paragraph does not explicitly mention "twenty study questions", so the remaining questions are formulated based on the content:
"Linear systems are the basis and a fundamental part of linear algebra, a subject used in most modern mathematics."
"They play a prominent role in engineering, physics, chemistry, computer science, and economics."
"A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system."
"The coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers."
"Integer linear programming is a collection of methods for finding the 'best' integer solution (when there are many)."
"For solutions in an integral domain like the ring of integers, or in other algebraic structures, other theories have been developed."