3 Systems of Linear Equations Free Online FlashCards

What is a system of linear equations?

A system of linear equations is a collection of linear equations considered simultaneously. Its solution is an ordered tuple satisfying every equation.

What does $A\mathbf{x}=\mathbf{b}$ represent?

In matrix form, a system is written as $A\mathbf{x}=\mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ contains the unknowns, and $\mathbf{b}$ contains the constants.

What is an augmented matrix?

The augmented matrix $[A\mid\mathbf{b}]$ joins the coefficient matrix and the constants column. Its vertical bar separates these columns; it is not an arithmetic operation.

Which row operations preserve a system’s solution set?

The three elementary row operations are row interchange, multiplication of a row by a nonzero constant, and adding a multiple of one row to another.

What conditions define row-echelon form?

In row-echelon form, zero rows are last, each leading entry is to the right of the one above it, and entries below each leading entry are zero.

How does RREF differ from REF?

Reduced row-echelon form additionally requires every pivot to equal $1$ and to be the only nonzero entry in its column. Every matrix has a unique RREF.

How do Gaussian and Gauss–Jordan elimination differ?

Gaussian elimination stops at row-echelon form and uses back-substitution. Gauss–Jordan elimination continues to reduced row-echelon form, where solutions can usually be read directly.

What distinguishes pivot and free variables?

A pivot variable corresponds to a pivot column. A free variable corresponds to a nonpivot column and can be assigned a parameter when the system is consistent.

How are solutions parameterized when $z$ is free?

For the displayed system, let $z=t$. Then $x=5-2t$ and $y=1+3t$, so $(x,y,z)=(5-2t,1+3t,t)$, where $t\in\mathbb{R}$.

What does a contradictory augmented-matrix row mean?

A row $[0\ 0\ \cdots\ 0\mid c]$ with $c\neq0$ represents $0=c$, a contradiction. Therefore, the system has no solution.

What are the three possible solution types?

A linear system has exactly one solution, infinitely many solutions, or no solution. A consistent system is unique when every variable is a pivot variable; a free variable gives infinitely many solutions.

How does rank classify a consistent system?

For a consistent system with coefficient matrix $A$, rank $n$ gives a unique solution when $A$ has $n$ columns. Rank less than $n$ gives infinitely many solutions.