MTH 0 Linear Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Week # Section.,. Sec... Systems of Linear Equations... D examples Example Consider a system of two equations with two unknowns x y = x + y = () There are two commonly used methods to solve linear systems elimination method and substitution method. We now use the elimination method by adding both equation to obtain y = Substitute into the rst equation of system (), we nd x = y = A solution of system () is an ordered couple (x 0 ; y 0 ) that satis es both equations. Here, the system has exact one solution (; ) Example Consider another system x y = x + y = () Adding both equations leads to 0 = This contradictory equation indicates that system () cannot possibly have a solution. Example Consider another system x y = x + y = ()
The second equation is a multiple of the rst equation (by -). Therefore, this system has only one independent equation. Consequently, there are in nitely many solutions for any choice of t x = y = t + t Geometrically, in D, the graph of a linear equation is a straight line. A solution of a system of two equation is an intersection of two lines. There are tow possible situations for two lines (a) two lines intersect each other (one solution), (b) two lines are parallel (no solution), and (c) two line are identical (any point on the line is a solution). This observation and the elimination method extends to general situations.... General systems. A system of m equations with n variables, denoted by x ; x ; ; x n ; reads as > > a x + a x + + a n x n = b a x + a x + + a n x n = b a m x + a m x + + a mn x n = b m () a ij is called a coe cient. It is in the ith equation and is associated with x j A solution of system () consists of n ordered numbers (x ; x ; ; x n ) satisfying all m equations. The set of all possible solutions is called a solution set. Example A system of equations with unknowns x x + x = 0 x x = x + x + 9x = 9 We use the same method of elimination Equation # is replaced by the sum of itself and times Equation # (while maintain the other equations) x x + x = 0 x x = x + x = 9 Next, We add Equation # to # to arrive at x x + x = 0 x x = x = () (6) We solve x from the last equation x = Substitute it into Equation #, we nd x = =) x = =
Finally, substitute x = ; x = into the rst equation x + = 0 =) x = 6 This whole process can be simpli ed using symbolic means. We introduce the coe cient matrix for system () a a a n 6 a a a n () a m a m a mn mn It has m rows and n columns. it is also called a m n matrix. The entire information of system () can be found in a a a n j b 6 a a a n j b j () a m a m a mn j b m It is called Augmented matrix. This is a m (n + ) matrix. The elimination method basically consists of the following row operations De nition The following operations are called elementary row operations. Replace one row by the sum of itself and a multiple of another row;. Interchange two rows;. One row is replaced by a non-zero multiple of itself. Theorem 6 Elementary row reductions do not alter the solution set of any system of linear equations. Example Solve system () using row operations. Solution Augmented matrix j 0 0 j 9 j 9 We now perform a series of row operation in the way equivalent to what we did before 0 j j 0 9 j 9 R + R R R + R R j 0 0 j 0 0 j j 0 0 j 0 j 9
The corresponding system is x x + x = 0 x x = x = which is exactly the same as before. We can continue row operations j 0 j 0 0 j 6 R = R 0 j 0 0 j 0 0 j j 0 j 0 R + R R 6 0 0 j R = R 6 0 0 j 0 0 j 0 0 j 6 6 0 j 0 0 j R + R R 6 0 0 j ( ) R + + R R 6 0 0 j 0 0 j 0 0 j 6 6 0 0 j = x 6 0 0 j (Reduced Echelon Form) =) x = 6 x 0 0 j Example Solve x x = x x + x = x x + x = Solution Write down the Matrix form and perform row operations 0 j j R R j 0 j j j R = R = j = 0 j j ( =) R + R R ( ) R + R R = j = 0 j 0 0 0 j = (9) = j = 0 j 0 = j = =
We convert back to the system > > x x + x = x x = 0 = =) impossible, means no solution.
Section.. Row reductions, Echelon forms... Echelon Forms The rst entry of a row (or column) is called the leading entry of the row (or column). De nition 9 A matrix is called in Echelon form (upper triangle form) if (a) All non-zero rows are above any zero-row (row with all entries zero), (b) For any two rows, the column containing the leading entry of the upper row is on the left of the column containing the leading entry of the lower row. Echelon form 0 0 0 0 0 0 0 0 non-echelon form ; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 De nition 0 If, in addition to (a) and (b) above, (c) all leading entries are and it is the only non-zero entry in the column, the matrix is called Reduced Echelon form. Reduced Echelon form 0 0 0 0 0 0 0 0 0 0 ; 0 0 0 0 0 0 0 0 0 Theorem Any matrix can be reduced by elementary row operations to the unique reduced Echelon form. The solution set of a system is the same as that of the system from the reduced Echelon form of the augmented matrix. De nition All leading entries are called pivot positions. All columns containing pivot positions are called pivot columns. The example in the end of last section can be summed up as... Gauss-Jordan Algorithm Gauss-Jordan Algorithm of nding reduced Echelon form Step. From the left, nd the rst non-zero column (it is a pivot column). By interchanging two rows if necessary, make sure that the leading entry of the column is non-zero. We end up with the following matrix (* represents any number) 0 a 6 0 b ; a 6= 0 0 c 6
Step. Perform elementary row operation # R =a R ; we arrive at 0 6 0 b 0 c Then, perform elementary row operation # several times (i.e., rst, R br R ;..., nally R m cr R m ), we obtain a matrix 0 6 0 0 A B 0 0 C Step. repeat the above two steps for the submatrix obtained from deleting the rst row 0 0 A B 0 0 C The matrix is then reduced row-equivalently to 0 W 6 0 0 0 0 0 By perform row operation R 6 W R R ; we have 0 0 0 0 0 0 0 Step. Repeat till reduced Echelon form. Example Find the reduced Echelon form for 0 j j 0 9 j 9 and then solve () x x + x = 0 x x = x + x + 9x = 9
Solution We perform row operations as follows j 0 0 j R + R R j 0 0 j 9 j 9 0 j 9 R = R j 0 0 j R + R R 0 j 9 R + R R 0 j 0 j R + R R 0 0 j R + R R 0 0 j 9 0 0 j 6 0 0 j Solution x = 9; x = 6; x = 0 j 0 j 0 j 9 0 0 j 9 0 j 0 0 j Theorem (Existence & Uniqueness) A system is called consistent if it admits at least one solution. A system is consistent if and only if the rightmost column contains NO pivot. in other words, there is no row having the form [0,0,...,0,b] (b 6= 0) Example In the second example of the previous section, we have 0 j = j = j 0 j j 0 0 0 j = The last column is a pivot column. Thus, it is inconsistent. Example 6 Determine if the system Sol. The augmented matrix Row operations 0 6 6 j j 9 9 9 6 j R R R Answer consistent. x 6x + 6x + x = x x + x x + x = 9 x 9x + x 9x + 6x = 0 6 6 j j 9 9 9 6 j R R j 9 0 6 6 j 0 j 6 j 9 0 6 6 j 9 9 6 j R + R R j 9 0 6 6 j 0 0 0 0 = j
Homework Section. #,,,,, Section. #,,,, 9, 9