Solving System of Linear Equations
Explore the concept of solving systems of linear equations with equivalent transformations and elementary row operations. Learn strategies to simplify complex systems and find solutions efficiently using matrices and augmented matrices.
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Presentation Transcript
Solving Solving System of System of Linear Equations Linear Equations
Equivalent Two systems of linear equations are equivalent if they have exactly the same solution set. 3?1 ?1 +?2 3?2 = 10 = 0 ?1 = 3 = 1 ?2 3 1 equivalent 3 1 Solution set: Solution set:
Equivalent Applying the following three operations on a system of linear equations will produce an equivalent one. 1. Interchange 3?1 ?1 3?2 = 0 ?1 3?1 3?2 +?2 = 0 = 10 +?2 = 10 2. Scaling (non zero) 3?1 3?1 +?2 +9?2 = 10 = 0 3?1 ?1 +?2 3?2 = 10 = 0 X(-3) 3. Row Addition 3?1 ?1 +?2 3?2 = 10 = 0X(-3) 10?2 3?2 = 10 = 0 ?1
Solving system of linear equation Strategy We know how to transform the given system of linear equations into another equivalent one. We do it again and again until the system of linear equation is very simple Finally, we know the answer at a glance. ?1 3?2 10?2 = 0 = 10 ?1 3?1 3?2 +?2 = 0 = 10 X 3 X 1/10 ?1 = 3 = 1 ?1 3?2 ?2 = 0 = 1 ?2 X 3
Augmented Matrix a system of linear equation ?11 ?21 ??1 ?12 ?22 ??2 ?1? ?2? ??? ?1 ?2 ?? ?1 ?2 ?? ? = m x n ? = ? = coefficient matrix
Augmented Matrix a system of linear equation m x (n+1) m x n m x 1 augmented matrix
Back to Equivalent 1. Interchange ?1 3?1 3?2 +?2 = 0 = 10 3?1 ?1 +?2 3?2 = 10 = 0 2. Scaling (non zero) 3?1 3?1 +?2 +9?2 = 10 = 0 3?1 ?1 +?2 3?2 = 10 = 0 X(-3) 3. Row Addition 3?1 ?1 +?2 3?2 = 10 = 0X(-3) 10?2 3?2 = 10 = 0 ?1
elementary row operations Back to Equivalent 1. Interchange Interchange any two rows of the matrix 1 3 3 1 0 10 3 1 1 10 0 3 Multiply every entry of some row by the same nonzero scalar 2. Scaling (non zero) 3 1 9 10 0 3 1 1 10 0 3 X(-3) 3 Add a multiple of one row of the matrix to another row 3. Row Addition 0 1 10 3 10 0 3 1 1 10 0 X(-3) 3
Solving system of linear equation A simple system of linear equations A complex system of linear equations R x = b Ax = b equivalent R=[ R b ] A =[ Ab ] A A elementary row operations 1. Interchange any two rows of the matrix 2. Multiply every entry of some row by the same nonzero scalar 3. Add a multiple of one row of the matrix to another row
X 3 ?1 3?1 3?2 +?2 = 0 = 10 X 3 1 3 3 1 0 10 ?1 3?2 10?2 = 0 = 10 X 1/10 1 0 3 10 0 10 X 1/10 ?1 3?2 ?2 = 0 = 1 1 0 3 1 0 1 X 3 X 3 ?1 = 3 = 1 1 0 0 1 3 1 ?2
Solving system of linear equation A simple system of linear equations A complex system of linear equations ????? R x = b Ax = b equivalent R=[ R b ] A =[ Ab ] A A Reduced Row Echelon Form (RREF) elementary row operations: 1. Interchange any two rows of the matrix 2. Multiply every entry of some row by the same nonzero scalar 3. Add a multiple of one row of the matrix to another row
