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Solving Systems of Linear Equations
We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in solve this problem.
Solving Systems of Linear Equations
We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in solve this problem. When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.
- How many solutions does the system of equations $3x + 2y = 4$, $6x + 4y = 8$ have? Explain your answer.
- How many solutions does the system of equations $3x + 2y = 4$, $6x + 4y = -2$ have? Explain your answer.
- What do we mean when we say mathematics is a language?
Systems of Linear Equations
Our first example is of a type we will not pursue further. While it has two equations, the first is not linear. So this is a good example to come back to later, especially after you have seen .
Example: Solving two (nonlinear) equations
Suppose we desire the simultaneous solutions of the two equations, x^2+y^2&=1\\ -x+\sqrt{3}y&=0\\
You can easily check by substitution that $x=\tfrac{\sqrt{3}}{2},\;y=\tfrac{1}{2}$ and $x=-\tfrac{\sqrt{3}}{2},\;y=-\tfrac{1}{2}$ are both solutions. We need to also convince ourselves that these are the only solutions. To see this, plot each equation on the $xy$-plane, which means to plot $(x,\,y)$ pairs that make an individual equation true. In this case we get a circle centered at the origin with radius 1 and a straight line through the origin with slope $\tfrac{1}{\sqrt{3}}$. The intersections of these two curves are our desired simultaneous solutions, and so we believe from our plot that the two solutions we know already are indeed the only ones. We like to write solutions as sets, so in this case we write the set of solutions as S&=\set{\left(\tfrac{\sqrt{3}}{2},\,\tfrac{1}{2}\right),\,\left(-\tfrac{\sqrt{3}}{2},\,-\tfrac{1}{2}\right)}
In order to discuss systems of linear equations carefully, we need a precise definition. And before we do that, we will introduce our periodic discussions about Proof Techniques. Linear algebra is an excellent setting for learning how to read, understand and formulate proofs. But this is a difficult step in your development as a mathematician, so we have included a series of short essays containing advice and explanations to help you along. These will be referenced in the text as needed, and are also collected as a list you can consult when you want to return to re-read them. (Which is strongly encouraged!)
With a definition next, now is the time for the first of our proof techniques. So study . We'll be right here when you get back. See you in a bit.
Now we make the notion of a solution to a linear system precise.
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Possibilities for Solution Sets
The next example illustrates the possibilities for the solution set of a system of linear equations. We will not be too formal here, and the necessary theorems to back up our claims will come in subsequent sections. So read for feeling and come back later to revisit this example.
Example: Three typical systems
Consider the system of two equations with two variables, 2x_1+3x_2&=3\\ x_1-x_2&=4
If we plot the solutions to each of these equations separately on the $x_{1}x_{2}$-plane, we get two lines, one with negative slope, the other with positive slope. They have exactly one point in common, $(x_1,\,x_2)=(3,\,-1)$, which is the solution $x_1=3$, $x_2=-1$. From the geometry, we believe that this is the only solution to the system of equations, and so we say it is unique.
Now adjust the system with a different second equation, 2x_1+3x_2&=3\\ 4x_1+6x_2&=6
A plot of the solutions to these equations individually results in two lines, one on top of the other! There are infinitely many pairs of points that make both equations true. We will learn shortly how to describe this infinite solution set precisely (see , ). Notice now how the second equation is just a multiple of the first.
One more minor adjustment provides a third system of linear equations, 2x_1+3x_2&=3\\ 4x_1+6x_2&=10
A plot now reveals two lines with identical slopes, parallel lines. They have no points in common, and so the system has a solution set that is empty, $S=\emptyset$.
This example exhibits all of the typical behaviors of a system of equations. A subsequent theorem will tell us that every system of linear equations has a solution set that is empty, contains a single solution or contains infinitely many solutions (). yielded exactly two solutions, but this does not contradict the forthcoming theorem. The equations in are not linear because they do not match the form of , and so we cannot apply in this case.
Equivalent Systems and Equation Operations
With all this talk about finding solution sets for systems of linear equations, you might be ready to begin learning how to find these solution sets yourself. We begin with our first definition that takes a common word and gives it a very precise meaning in the context of systems of linear equations.
Notice here that the two systems of equations could look very different ( not be equal), but still have equal solution sets, and we would then call the systems equivalent. Two linear equations in two variables might be plotted as two lines that intersect in a single point. A different system, with three equations in two variables might have a plot that is three lines, all intersecting at a common point, with this common point identical to the intersection point for the first system. By our definition, we could then say these two very different looking systems of equations are equivalent, since they have identical solution sets. It is really like a weaker form of equality, where we allow the systems to be different in some respects, but we use the term equivalent to highlight the situation when their solution sets are equal.
With this definition, we can begin to describe our strategy for solving linear systems. Given a system of linear equations that looks difficult to solve, we would like to have an equivalent system that is easy to solve. Since the systems will have equal solution sets, we can solve the easy system and get the solution set to the difficult system. Here come the tools for making this strategy viable.
These descriptions might seem a bit vague, but the proof or the examples that follow should make it clear what is meant by each. We will shortly prove a key theorem about equation operations and solutions to linear systems of equations.
We are about to give a rather involved proof, so a discussion about just what a theorem really is would be timely. Stop and read first.
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Symbols used here
The non-negative number whose square (n-th power) is x.
i² = −1.
Inequalities that allow equality; < and > exclude it.
A quantity with magnitude and direction; a column of numbers.
A rectangular array of numbers; a linear map.
Scaling factor of area/volume under A; zero means singular.
The matrix that undoes A; A with rows and columns swapped.
The factor by which an eigenvector is stretched: Av = λv.
Σ u_i v_i; the length of v, √(v·v).
Questions people ask
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn — it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
Sankhani wanu
Parts of this page are adapted from Beezer, A First Course in Linear Algebra (GFDL 1.2). Condensed and re-explained here; errors are ours.
Zambiri pa Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis