This page presents some topics from Linear Algebra needed for construction of solutions to systems of linear algebraic equations and some applications. We use matrices and vectors as essential elements in obtaining and expressing the solutions.
The central problem of linear algebra is to solve a linear system of equations. This means that the unknowns are only multiplied by numbers.
Our first example of a linear system has two equations in two unknowns:We begin with a row at a time. The first equation \( x-3y=-1 \) produces a straight line in the \( xy- \) plane. The second equation \( 2x+y=5 \) defines another line. By plotting these two lines, you cannot miss the intersection point where two lines meet. The point \( x=2, \ y=1 \) lies on both lines. That point solves both equations at once.
Now we visualize this with a picture: