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Unformatted text preview: MA 35100 LECTURE NOTES: MIDTERM #1 REVIEW 1.1: Introduction to Linear Systems Geometric Interpretation. A system of linear equations is the intersection of hyperplanes. For example, the equation x + 2 y + 3 z = 39 represents a plane in R 3 , so the three equations represent the intersection of three planes. A system may either have one solution, infinitely many solutions, or no solutions. 1.2: Matrices, Vectors, and Gauss-Jordan Elimination Matrices. Definition: An n m matrix is a rectangular array of n rows and m columns: A = a 11 a 12 ... a 1 m a 21 a 22 ... a 2 m . . . . . . . . . . . . a n 1 a n 2 ... a nn . The entries of the matrix are denoted by a ij , where the first subscript i denotes the row and the second subscript j denotes the column. The entry a ij is located in the i th row and the j th column. We say that A is a square matrix if the number of rows equals the number of columns i.e. m = n . If A is a square matrix, the entries a 11 , a 22 , ..., a ii , ... form the main diagonal of A . Let A be a square matrix i.e. an n n matrix. If the entries below the main diagonal are zero i.e. a ij = 0 for i > j then we say A is an upper triangular matrix . Similarly, if the entries above the main diagonal are zero i.e. a ij = 0 for i < j then we say A is a lower triangular matrix . A square matrix is called diagonal if it is both upper triangular and lower triangular i.e. the entries above and below the main diagonal are zero. Vectors. Definition: Let A be an n m matrix. If A has only one column i.e. m = 1 then A is called a column vector . The set of all column vectors is denoted by R n . If A has only one row i.e. n = 1 then A is called a row vector . If A has only one column and one row i.e. m = n = 1 then A is called a scalar . The entries of a column vector or a row vector are called its components . Reduced Row-Echelon Form. The system of equations a 11 x + a 12 y + a 13 z = a 14 a 21 x + a 22 y + a 23 z = a 24 a 31 x + a 32 y + a 33 z = a 34 may be represented by the augmented matrix a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 . To solve such a system, we perform the following steps: Step 1: If the cursor entry is zero, swap the row with a row below having a nonzero entry in that column. Step 2: Divide the cursor row with the nonzero entry to make the nonzero entry equal to 1. 1 Step 3: Eliminate all other entries in the cursor column by subtracting suitable multiples of the cursor row. Step 4: Move the cursor down one row, and over to the right. If all the entries below are zero, continue to more to the right. The operations in Steps 1, 2, and 3 are called elementary row operations . The resulting matrix is called reduced row-echelon form (rref)....
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