Systems of Linear Equations and Row Reduction: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 1, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
An equation of the form a1 x1 + a2 x2 + ... + an xn = b, in which every unknown appears to the first power only and no two unknowns are multiplied together.
System of linear equations
A finite collection of linear equations in the same list of unknowns, considered simultaneously; a solution is an assignment of values satisfying every equation at once.
Solution set
The set of all ordered tuples that satisfy every equation of a system. Two systems are called equivalent exactly when they have the same solution set.
Consistent system
A system possessing at least one solution. Geometrically the associated hyperplanes have at least one common point of intersection.
Inconsistent system
A system with no solution at all. In row reduced form it is detected by a row reading 0 = c for some nonzero constant c.
Coefficient matrix
The rectangular array whose entry in row i and column j is the coefficient of the jth unknown in the ith equation, with the constants on the right omitted.
Augmented matrix
The coefficient matrix with the column of right hand constants attached, usually drawn after a vertical bar; it records the entire system with no loss of information.
Elementary row operation
One of three reversible moves on a matrix: swapping two rows, scaling a row by a nonzero constant, or adding a multiple of one row to another row.
Row equivalent matrices
Two matrices connected by a finite sequence of elementary row operations. Row equivalent augmented matrices always represent systems with identical solution sets.
Echelon form
A matrix shape in which all zero rows sit at the bottom and each leading nonzero entry lies strictly to the right of the leading entry in the row above it.
Reduced row echelon form
Echelon form with the extra demands that every leading entry equals one and is the only nonzero entry in its column. Each matrix has exactly one such form.
Uniqueness of the reduced form
A theorem stating that every matrix is row equivalent to one and only one reduced row echelon matrix, no matter which sequence of row operations is chosen.
Pivot position
A location in a matrix that holds a leading entry of the reduced row echelon form; the column containing it is called a pivot column.
Basic variable
An unknown whose column is a pivot column. In the general solution each basic variable is written explicitly in terms of the free variables.
Free variable
An unknown whose column carries no pivot. It may take any scalar value, and each free variable adds one parameter to the description of the solution set.
Gaussian elimination
The forward phase of row reduction: work down the matrix creating zeros beneath each pivot until echelon form is reached, then read the system by back substitution.
Gauss Jordan elimination
Row reduction continued past echelon form, clearing entries above each pivot and scaling pivots to one, so the solution can be read off with no back substitution.
Existence and uniqueness theorem
A system is consistent exactly when the rightmost column of the augmented matrix is not a pivot column; when consistent it has a unique solution precisely when there are no free variables.
Homogeneous system
A system whose constants are all zero, written A x = 0. It is always consistent because the zero tuple is a solution, called the trivial solution.
Nontrivial solution
A solution of a homogeneous system with at least one nonzero entry. Such a solution exists exactly when the system has at least one free variable.
Parametric vector form
A description of a solution set as a particular vector plus a linear combination of vectors scaled by the free variables, making the geometry of the set visible.
General solution structure
If A x = b is consistent, its solution set is one particular solution translated by the whole solution set of the matching homogeneous system A x = 0.
Overdetermined system
A system with more equations than unknowns. Such systems are frequently inconsistent, though extra equations may simply repeat information already present.
Underdetermined system
A system with fewer equations than unknowns. It always has free variables, so it is either inconsistent or has infinitely many solutions.
Vector equation view of a system
The statement that A x = b is solvable exactly when b can be written as a linear combination of the columns of A, with the entries of x as the weights.