Introduction to systems of linear equationsThese slides are based on Section 1 inLinear Algebra and its Applicationsby David C. Lay.Definition 1.Alinear equationin the variablesx1,...,xnis an equation that can bewritten asa1x1+a2x2+ +anxn=b.Example 2.Which of the following equations are linear?•4x1−5x2+2=x1•x2=2( 6√−x1)+x3•4x1−6x2=x1x2•x2=2x1√−7Definition 3.•Asystem of linear equations(or alinear system) is a collection of one or morelinear equations involving the same set of variables, say,x1,x2,...,xn.•Asolutionof a linear system is a list(s1,s2,...,sn)of numbers that makes eachequation in the system true when the valuess1,s2,...,snare substituted forx1,x2,...,xn, respectively.Example 4. (Two equations in two variables)In each case, sketch the set of all solutions.x1+x2= 1−x1+x2= 0x1−2x2=−32x1−4x2= 82x1+x2= 1−4x1−2x2=−2Theorem 5.A linear system has either•no solution, or•one unique solution, or•infinitely many solutions.Definition 6.A system isconsistentif a solution exists.Armin Straub[email protected]1
How to solve systems of linear equationsStrategy: replace system with an equivalent system which is easier to solveDefinition 7.Linear systems areequivalentif they have the same set of solutions.Example 8.To solve the first system from the previous example:x1+x2= 1−x1+x2= 0R2→R2+R1x1+x2= 12x2= 1Once in thistriangularform, we find the solutions byback-substitution:x2=1/2,x1=Example 9.The same approach works for more complicated systems.x1−2x2+x3=02x2−8x3=8−4x1+ 5x2+ 9x3=−9R3→R3+4R1x1−2x2+x3=02x2−8x3=8−3x2+13x3=−9R3→R3+32R2x1−2x2+x3= 02x2−8x3= 8x3= 3By back-substitution:x3=3,x2=,x1=It is always a good idea to check our answer. Let us check that(29,16,3)indeed solvesthe original system:x1−2x2+x3=02x2−8x3=8−4x1+ 5x2+ 9x3=−9Armin Straub[email protected]2
Matrix notationx1−2x2=−1−x1+ 3x2=3bracketleftbiggbracketrightbigg(coefficient matrix)bracketleftbiggbracketrightbigg(augmented matrix)Definition 10.Anelementary row operationis one of the following:•(replacement)Add one row to a multiple of another row.•(interchange)Interchange two rows.•(scaling)Multiply all entries in a row by a nonzero constant.