Linear algebraweek2
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The document discusses row operations and reduced row-echelon form. There are three types of row operations that can be performed on a matrix: swapping rows, multiplying a row by a nonzero number, and multiplying a row by a number and adding it to another row. Reduced row-echelon form has specific properties: rows of all zeros must be below non-zero rows, the leftmost non-zero entry in a row must be 1 and the only non-zero in its column, and leftmost non-zero entries must be in ascending order of row and column.
Linear algebraweek2
- 1. LINEAR ALGEBRA Associate professor Faiz ul haque Zeya Bahria University Karachi Campus
- 2. Row operation Definition RO Row Operations The following three operations will transform an m×nm×n matrix into a different matrix of the same size, and each is known as a row operation. Swap the locations of two rows. Multiply each entry of a single row by a nonzero quantity. Multiply each entry of one row by some quantity, and add these values to the entries in the same columns of a second row. Leave the first row the same after this operation, but replace the second row by the new values.
- 3. cont We will use a symbolic shorthand to describe these row operations: Ri↔RjRi↔Rj : Swap the location of rows ii and jj . αRiαRi : Multiply row ii by the nonzero scalar αα . αRi+RjαRi+Rj : Multiply row ii by the scalar αα and add to row jj .
- 4. RREF Reduced Row-Echelon Form Definition RREF Reduced Row-Echelon Form A matrix is in reduced row-echelon form if it meets all of the following conditions: If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry. The leftmost nonzero entry of a row is equal to 1. The leftmost nonzero entry of a row is the only nonzero entry in its column. Consider any two different leftmost nonzero entries, one located in row ii , column jj and the other located in row ss , column tt . If s>is>i , then t>jt>j .