SECTIDefinitio- A- C- M SECTION 1.1 SYons: A system of lConsistent anMatrix notatioON 1.2 RYSTEMS linear equatind inconsisteon; CoefficiRow ReducOF LINEons; Solutioent ent matrix; Action and EEAR EQUAn set; EquivAugmented mEchelon FATIONSvalence of twmatrix Forms wo systems ROW REDUCTION ALGORITHM - STEP 1: Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top. - STEP 2: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position. - STEP 3: Use row replacement operations to create zeros in all positions below the pivot. - STEP 4: Cover the row containing the pivot position, and cover all rows, if any, above it. Apply steps 1–3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify. - STEP 5: Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation. SECTION 1.3 VECTOR EQUATIONS - Two vectors are equal if and only if their corresponding entries are equal. - Given two vectors u and v, their sum is the vector obtained by adding corresponding entries of u and v. - Given a vector u and a real number c, the scalar multiple of u by c is the vector cu obtained by multiplying each entry in u by c. SECTION 1.4 THE MATRRIX EQUA ATION AAx = b Section 1- AA- S- T. 1.5 SOLUTIOA system of lAx=0, where uch a systemThis zero soluON SETS OF Llinear equatiA is an m Xm Ax=0 alwaution is usuaLINEAR SYSTons is said toX n matrix anays has at leally called thTEMS o be homognd 0 is the zeast one soluthe trivial solu eneous if it ero vector intion, namelyution can be writtn ܴ y, (the zero v ten in the forvector in ). rm
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