Systems and matrices · Definition

System of linear equations

linear algebrasystems

A linear system is a set of first-degree equations that must all hold at once; it is the problem from which all of linear algebra grows.

Definition

A system of linear equations is a set of linear equations in the same unknowns. A solution is an assignment of values that satisfies all the equations at once.

\[\begin{aligned}a_{11}x_1+\cdots+a_{1n}x_n&=b_1\\ \vdots\qquad&\\ a_{m1}x_1+\cdots+a_{mn}x_n&=b_m\end{aligned}\]

Linear systems appear long before anyone names them: whenever two conditions must hold at once, such as «there are 10 coins in total and their value adds up to 35 pesos», each condition gives an equation and there are several unknowns. Linear algebra grows out of the wish to solve such problems systematically, without relying on cleverness in each case.

It is called linear because each unknown appears only to the first power, multiplied by a number; there are no products of unknowns, no squares, no functions such as \(\sin x\). That restriction is exactly what makes a complete theory possible: in the plane each equation \(ax+by=c\) is a line, and solving the system asks where the lines meet. Two lines may cross at one point (a unique solution), coincide (infinitely many solutions) or be parallel (none).

A guiding example: \(x+y=3\) and \(x-y=1\) have the unique solution \((2,1)\), but \(x+y=3\) and \(2x+2y=6\) describe the same line, and \(x+y=3\) with \(x+y=5\) are inconsistent. A typical mistake is to believe that more equations than unknowns means no solution, or the opposite; what matters is whether the equations are independent and consistent. This concept opens the course: matrices and Gaussian elimination are the tools for studying it.

Think about it

Can a linear system have exactly two solutions?

No. If it has two distinct solutions \(\mathbf x\) and \(\mathbf y\), every combination \(t\mathbf x+(1-t)\mathbf y\) is also a solution, so there are infinitely many. The only options are zero, one or infinitely many solutions.

Exercises 10

  1. Which of the following equations is linear in the unknowns \(x\) and \(y\)?
  2. Solve the system \(x+y=5,\; x-y=1\).
  3. Which of the following systems has \((x,y)=(2,1)\) as a solution?
  4. How many solutions does the system \(x+y=2,\; 2x+2y=4\) have?
  5. How many solutions does the system \(x+y=1,\; x+y=3\) have?
  6. For which value of \(k\) does the system \(x+y=1,\; 2x+2y=k\) have infinitely many solutions?
  7. Solve the system \(x+y+z=6,\; x-y+z=2,\; 2x+y-z=1\).
  8. What is the augmented matrix of the system \(2x-y=3,\; x+4y=-1\)?
  9. For which values of \(k\) does the system \(x+2y=1,\; 3x+ky=2\) have a unique solution?
  10. About a homogeneous system (all constant terms equal to 0), which statement is always true?

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §2.1
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §1.A