Introduction to systems of linear equationsThese slides are based o n Section 1 in Linear Algebra and its Applications by David C. Lay.Definition 1. A linear equation in the v ariables x1, ..., xnis an equation that can bewritten asa1x1+ a2x2++ anxn= b.Example 2. Which of the following equations are linear?•4x1−5x2+ 2 = x1linear: 3x1−5x2= −2•x2= 2( 6√−x1) + x3linear: 2x1+ x2−x3= 2 6√•4x1−6x2= x1x2not linear: x1x2•x2= 2 x1√−7 not linear: x1√Definition 3.•A system of linear equations (or a linear s ystem) is a collection of one or morelinear equations involving the same set of v ariables, say, x1, x2, ..., xn.• A solution of a linear system is a list (s1, s2, ..., sn) of numbers that makes eachequation in the system true when the values s1, s2, ..., snare substituted for x1, x2,..., xn, respectively.Example 4. (Two equations in two variables)In each ca se, sketch the set of all s olutions.x1+ x2= 1−x1+ x2= 0x1− 2x2= −32x1− 4x2= 82x1+ x2= 1−4x1− 2x2= −2-3-2-1123-3-2-1123-3-2-1123-3-2-1123-3-2-1123-3-2-1123Theorem 5. A linear system has e ither• no soluti on, or• one unique solution, or• infinitely many solut ions.Definition 6. A system is consistent if a solution exists.Armin [email protected] to solve syst ems of linear equationsStrategy: repla c e system with an equivalent system which is easie r to solveDefinition 7. Linear systems are equivalent if they have the same set of solutions.Example 8. To solve the first system from the previous example:x1+ x2= 1−x1+ x2= 0>R2→R2+R1x1+ x2= 12x2= 1Once in this triangular form, we find the solutions by back-substitution:x2= 1/2, x1= 1/2Example 9. The same approach works f or mor e complic ated systems.x1− 2x2+ x3= 02x2− 8x3= 8−4x1+ 5x2+ 9x3= −9,R3 → R3 + 4R1x1− 2x2+ x3= 02x2− 8x3= 8− 3x2+ 13x3= −9,R3 → R3 +32R2x1− 2x2+ x3= 02x2− 8x3= 8x3= 3By back-substitution:x3= 3, x2= 16, x1= 29.It is always a good idea to check our answer. Let us check that (29, 16, 3) i ndeed solvesthe original system:x1− 2x2+ x3= 02x2− 8x3= 8−4x1+ 5x2+ 9x3= −929 − 2 ·16 + 3@02 ·16 − 8 ·3@8−4 ·29 + 5 ·16 + 9 ·3@−9Matrix notationx1− 2x2= −1−x1+ 3x2= 31 −2−1 3(coefficient matrix)1−2 −1−13 3(augmented matrix)Armin [email protected] 10. An elementary row operation is one of the following:• (replacement) Add one row to a multiple of anoth e r row.• (interchange) Interchange two rows.• (scaling) Multiply all entries in a row by a non ze r o constant.Definition 11. Two matrices are row equivalent, if one matrix can be transformedinto the other matrix by a sequence of elementary row operations.Theorem 12. If the augmented matrice s of two li near systems are row equivalent, thenthe two systems have the same solution set.Example 13. Here is t he previous example in matrix notation.x1− 2x2+ x3= 02x2− 8x3= 8−4x1+ 5x2+ 9x3= −91 −21 00 2 −8 8−4 5 9 −9,R3 →R3 + 4R1x1− 2x2+ x3= 02x2− 8x3= 8− 3x2+ 13x3= −91 − 21 00 2 −8 80 − 3 13 −9,R3 →R3 +32R2x1− 2x2+ x3= 02x2− 8x3= 8x3= 31 − 21 00 2 −8 80 0 1 3Instead of back-substitution, we can continue with row operations.AfterR2 →R2 + 8R3, R1 →R1 −R3, we obtain:x1− 2x2= −32x2= 32x3= 31 −2 0 −30 2 0 320 0 1 3Finally, R1 →R1 + R2, R2 →12R2 results in:x1= 29x2= 16x3= 31 0 0 290 1 0 160 0 1 3We again find the solution (x1, x2, x3) = (29, 16, 3).Armin [email protected] reduction and echelon formsDefinition 14. A matrix is in e chelon form (or row echelon form) if:(1) Each leading entry (i .e . leftmost nonze ro entry) of a row is in a column to the rightof the leading entry of the row above it.(2) All entries in a c olumn below a leading entry are zero.(3) All nonzero rows are above any rows of all zeros.Example 15. Here is a representa t ive matrix in echelon form.0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗0 0 0 0 0 0 0 ∗ ∗ ∗0 0 0 0 0 0 0 0 ∗ ∗0 0 0 0 0 0 0 0 0 0 0(∗ stands for any value, and for any nonzero value.)Example 16. Are the following matrices in echelon form?(a) ∗ ∗ ∗ ∗0 ∗ ∗ ∗0 0 0 0 00 0 0 0 0YES(b)0 ∗ ∗ ∗ ∗ ∗ ∗ ∗0 0 0 0 00 0 0 0 0NOPE (b ut it is after exchanging th e first two rows)(c) ∗ ∗0 ∗0 0 0 0 0YES(d) 0 0∗ 0∗ 0 ∗ 0 0NORelated and extra material• In our textbook: parts of 1.1, 1.3, 2.2 (just pages 78 and 79 )However, I would suggest waiting a bit bef ore re ading through these parts (say, until we coveredthings like matrix multiplication in class).• Suggested practice exerc ise: 1, 4, 5, 10, 11 from Section 1.3Armin
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