Pre-lecture triviaWho are these four?• Artur Avila, Manjul Bhargava, Martin Hairer, Maryam Mirzakhani• Just won the Fields Medal!◦ analog to Nobel prize in mathematics◦ awarded every four years◦ winners have to be younger than 40◦ cash prize: 15,000 C$Ar min [email protected]• Each linear system corresponds to an augmented matrix.2x1−x2= 6−x1+2x2−x3= −9−x2+2x3= 122 −16−1 2 −1 −9−1 2 12augmented matrix• To solve a system, we perform row reduction.>R2→ R2+12R12 −1 0 6032−1 − 60 −12 12>R3→ R3+23R22 −10 6032−1 − 60 0438echelon form!• Echelon form in general: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 0The leading terms in each row are the pivots.Ar min [email protected] reduction and e chelon forms, continuedDefinition 1. A matrix is in r educed echelon form if, in addition to being in echelonform, it also satisfies:• Each pivot is 1.• Each pivot is the only n onzero entry in its column.Example 2. Our initial matrix in echelon form put into reduced 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 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 0 0 0 0 0 0 0 0 0 0Note that, to be in reduced echelon form, the pivots also have to be scaled to 1.Example 3. Are the following matrices in reduced echelon form?(a)0 1 ∗ 0 0 ∗ ∗ 0 0 ∗ ∗0 0 0 1 0 ∗ ∗ 0 0 ∗ ∗0 0 0 0 1 ∗ ∗ 0 0 ∗ ∗0 0 0 0 0 0 0 1 0 ∗ ∗0 0 0 0 0 0 0 0 1 ∗ ∗YES(b)1 0 5 0 −70 2 4 0 −60 0 0 −5 00 0 0 0 0NO(c)1 0 −2 3 2 −240 1 −2 2 0 −70 0 0 0 1 4NOTheorem 4. (Uniqueness of th e reduced ec helon form) Each matrix is row equiv-alent to one and only one redu c e d echelon matrix.Question. Is the sam e statement true for the echelon form?Clearly not; for instance,1 20 1 1 00 1are different row equivale nt ec helon forms.Ar min [email protected] 5. Row reduce to echelon form (often called Gaussian elimination) and thento reduced echelon form (often called Gauss–Jordan elimin ation):0 3 −6 6 4 −53 −7 8 −5 8 93 −9 12 −9 6 15Solution.After R1 ↔ R3, we g e t: (R1 ↔ R2 would be another option; try it!)3 −9 12 −9 6 153 −7 8 −5 8 90 3 −6 6 4 −5Then, R2 → R2 − R1 yields:3 −9 12 −9 6 150 2 −4 4 2 −60 3 −6 6 4 −5Finally, R3 → R3 −32R2 pro duces the echelon form:3 −9 12 −9 6 150 2 −4 4 2 −60 0 0 0 1 4To get the reduced echelon form, we first scale all rows:1 −3 4 −3 2 50 1 −2 2 1 −30 0 0 0 1 4Then, R2 → R2 − R3 and R1 → R1 − 2R3, gives:1 −3 4 −3 0 −30 1 −2 2 0 −70 0 0 0 1 4Finally, R1 → R1 + 3R2 produces the reduced echelon form:1 0 −2 3 0 − 2 40 1 −2 2 0 −70 0 0 0 1 4Ar min [email protected] of li near systems via row re ductionAfter row reduction to echelon form, we can easily solve a linear system.(especially after reduction to reduced echelon form)Example 6.1 6 0 3 0 00 0 1 −8 0 50 0 0 0 1 7 x1+6x2+3x4= 0x3−8x4= 5x5= 7• The pivots are located in columns 1, 3, 5. The correspondin g variables x1, x3, x5arecalled pivot variables (or ba sic variables).• The remaining variables x2, x4are called free variables.• We can solve each equation for the pi vot variables i n ter ms of the free variables (ifany). Here, we get:x1+6x2+3x4= 0x3−8x4= 5x5= 7x1= −6x2− 3x4x2freex3= 5 + 8x4x4freex5= 7• This is the gene ral solu tion of this system. The solution is in parametric form, withparameters given by the free variables.• Just to make sure: Is the above system consistent? Does it have a unique solution?Example 7. Find a parametric description of the solution set of:3x2−6x3+6x4+4x5= −53x1−7x2+8x3−5x4+8x5= 93x1−9x2+12x3−9x4+6x5= 15Solution. The augmented matrix is0 3 −6 64 −53 −7 8 −5 8 93 −9 12 −9 6 15.We determined earlier that its reduced echelon form is1 0 −2 3 0 −240 1 −2 20 −70 0 0 0 1 4.Ar min [email protected] pivot variables are x1, x2, x5.The free variables are x3, x4.Hence, we find the general solution as:x1= −24 + 2x3− 3x4x2= −7 + 2x3− 2x4x3freex4freex5= 4Related and extra material• In our textbook: still, parts of 1.1, 1.3, 2.2 (just pa g es 78 and 79)As before, I would suggest waiting a bit before r eading through these parts (say, until we cov eredthings like matrix multiplication in class).• Suggested practice exercise:Section 1.3: 13, 20; Section 2.2: 2 (only reduceA, B to echelon form)Ar min
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