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plugin-Week1Syllabus - Week 1.5 Syllabus: Systems of Linear...

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(1) System of linear equations : m linear equations n j =1 a ij x j = b i for i = 1 ,...,m in n unknowns x 1 ,...,x n ; called homogeneous if all b i = 0. A = [ a ij ] is the matrix of coefficients , [ A k b ] is the augmented matrix obtained by adjoining the stub b = b 1 . . . b m . For homogeneous systems we usually omit the stub (it remains constantly zero throughout the reduction process). A solution of a system is a value ( x 1 ,...,x n ) = ( a 1 ,...,a n ) for the variables which satisfies all equations of the system simultaneously . A system either has no solutions ( inconsistent system ), a single solution ( unique system ) [ independent of any further parameters], or has infinitely many solutions ( infinite system [ dependent on a choice of free parameters]. IT IS RECOMMENDED THAT YOU ALWAYS WRITE THE VARIABLES ABOVE THEIR COLUMNS IN THE MATRIX OF COEFFICIENTS A in a system AX = B , AND SEPARATE THE STUB B FROM THE MATRIX OF COEFFI- CIENTS BY A DOUBLE BAR k (reminding you of an upright elongated equal sign), for example x 1 x 2 x 3 2 - 1 3 5 6 8 2 9 3 0 5 7 . The augmented matrix may have a generic stub whose entries are variables b i or y i instead of constants (when you are solving Ax = y for all possible y ’s at once), or it may contain several numerical columns b i (when you are simultaneously solving several systems Ax = b i with the same A ). SKILL : be able to translate back and forth between a system of explicit equations and its augmented matrix. A shmoo system presents the answer to you directly (the augmented matrix is in reduced row-echelon form). (2) Elementary row operations E on matrices: (I) E ( i,j ) swaps (interchanges) the i th and j th rows, (II) E ( i,c ) scales or multiplies the i th row by nonzero constant c , (III) E ( i,c,j ) ( addmul ) multiplies the j th row by c and adds it to the i th . Fact: The elementary operations are reversible: the reverse of swap is swap, the reverse of scale by c is scale by the reciprocal 1 /c , the reverse of addmul by c is addmul by the negative - c (add - c times the original row to the altered one). Thus these operations on rows do not change the mathematical content of the system (the solution set), they merely changes the packaging. [Beware: the analogous elementary operation on columns DO change the mathematical content, since they change the variables; the variables are encoded in the augmented matrix by positional notation as labels for the columns, and these must not be mixed or altered. This is the reason it is useful to label the columns with their variables, which labeling stays the same throughout the reduction process.] (3) Echelon and Reduced row-echelon form : A matrix is in [row]echelon form (ref) iff (i) all zero rows are at the bottom, (ii) all nonzero rows begin with a leading one , (iii) the leading ones appear in echelon (stair-step) form, descending downward and to the right. This implies (iv) 0 the entries below any leading one are all zero (the following rows are
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plugin-Week1Syllabus - Week 1.5 Syllabus: Systems of Linear...

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