Linear algebra in turn is built on two basic elements, MATRICES and VECTORS.

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1 M-Lecture():.-. Linear algebra provides concepts that are crucial to man areas of information technolog and computing, including: Graphics Image processing Crptograph Machine learning Computer vision Optimiation Graph algorithms Quantum computation Computational biolog Information retrieval and web search Linear algebra in turn is built on two basic elements, MATRICS and VCTORS.

2 M-Lecture():.-.. Sstem of Linear quations Sstem of Linear quations: A sstem of m linear equations in n variables, or linear sstem, is a collection of equations of the form: This is also referred to as an m n linear sstem. If b =b =..=b m =, then we sa that the sstem is homogenous. For eample, the collection of equations Is a linear sstem of three equations in four variables, or linear sstem. A solution to a linear sstem with n variables is an ordered sequence ( s, s,..., sn ). ample: Solve the linear sstem Solution

3 M-Lecture():.-. The limination Method: The elimination method, also called Gaussian elimination, is an algorithm used to solve linear sstem. To describe this algorithm, we first introduce the triangular form of linear sstem. An m n linear sstem is in triangular form is when the coefficient a ij = whenever i j. Two eamples of triangular sstems are: and When a linear sstem is in triangular form, then the solution set can be obtained using a technique called back substitution. A sstem of equations is consistent if there is at least one solution to the sstem. If there are no solutions, the sstem is inconsistent.

4 M-Lecture():.-. ample: Consider the linear sstem given b Solution : From the last equation we get that =. Put this in equation (), this will give that : -()=, so =. Finall, using these values in the first equation, we have -()+=-, so =9 The solution is written as (9,,). ample: Consider the linear sstem given b Solution: First, we convert the sstem to an equivalent triangular sstem We eliminate the variable in the second and third equations: Interchange the second and the third equations gives the triangular sstem: Using back substitution, we have =,= and =--=. Therefore, the sstem in consistent with the unique solution (,,). Definition: Two linear sstems are equivalent if the have the same solution...

5 M-Lecture():.-. ample: Solve the linear sstem 8 Solution: Since ever term of the third equation can be divided evenl b, we get that : 8 8 Interchange and ( this is because we want the first equation to has with as coefficient) 8 We want to eliminate from the second and the third equations, to do that we use the operations Which is an equivalent sstem in triangular form. Using back situation, the general solution is,, With free to assume an real number. It is common in this case to replace with the parameter t. The general solution can be written as: S= {( t-,t-,t+,t) /t R}, and is called a one-parameter famil of solutions. A particular solution can be obtained b letting t be a specific value. For eample, if t=, then the particular solution is (-,-,,)... Read amples, & in the book pages: 9,,

6 M-Lecture():.-.. Matrices and lementar Row Operations In this section we will introduce matrices to help representing linear sstem. Then solution of a linear sstem is obtained b performing appropriate operations on a certain matri Definition: - Matri An m n matri is an arra of numbers with m rows and n columns. For eample, the arra numbers -Augmented Matri and Coefficient Matri Notice that for an m n linear sstem the augmented matri is m (n+).

7 M-Lecture():.-. amples 7

8 M-Lecture():.-. ample: chelon Form of a Matri Definition: An m n matri A is said to be in reduced row echelon form if it satisfies the following conditions: All ero rows (consisting entirel of eros), if an, are at the bottom. The first nonero entr from the left of a nonero row is, called the leading for that row. ach leading is to the right of all leading s in the rows above it. ach leading is to the onl nonero entr in its column. 8

9 M-Lecture():.-. 9 ample: Definition: The elementar row operations on an m n matri A are: Interchanging two rows. Multipling one row b a nonero number. Add a multiple of one row to a different row. The matri B is row equivalent to the matrices A.. e.g. B R R R R R R A

10 M-Lecture():.-. The process of transforming a matri to reduced row echelon form is called Gauss-Jordan elimination. ample: Find a matri B in reduced to the matri Solution: A row echelon form that is row equivalent

11 M-Lecture():.-. Let AX = B and CX = D be two sstems of linear equations each of m equations in n unknowns. If the augmented matrices [A B ] and [C D ] of these sstems are row equivalent, then both linear sstems have eactl the same solutions. To solve the sstem AX = B : Form the augmented matri [A B ]. Find the matri [C D ] in reduced row echelon form that is row equivalent to the matri [A B ] that b using elementar row operations. For the matri [C D ], there are possibilities: o Number of leading s = number of unknowns (variables), then the sstem has the unique solution X = D. o Number of leading s < number of unknowns, then the sstem has infinitel man solutions. Here the non-leading variables (unknowns corresponding to columns that do not contain leading ) end up as parameters and the leading variables (unknowns corresponding to columns that contain leading ) are given in terms of these parameters. o The sstem is inconsistent ( =!!!), the sstem has no solution.

12 M-Lecture():.-. ample: Solve the linear sstem b transforming the augmented matri to reduced row echelon form. Solution: The augmented matri of the linear sstem is We want to transform the matri into reduced row echelon form, we use the leading in row (as pivot) to eliminate the terms in column of rows, and. To do this we use the three operations: this will change to For the second step we use the left in row as the pivot and eliminate the two terms below the pivot. The operation we will use are: Reducing the matri to 8 -R +R R R +R R -R +R R R +R R -R +R R -R +R R

13 M-Lecture():.-. In row, to obtain a leading we divide each entr in row evenl b(-), which result is the matri: Now b using the leading in row as pivot, the operations Row- reduce the matri to 9 Using the operation R R, we change the signs of the entries in row to obtain the matri 9 Finall, using the leading in row as the pivot, we eliminate the terms above it in column. We use the operations: Applied to the last matri give 9 9 Which is in reduced row echelon form. R +R R -R +R R R +R R R +R R -R +R R R +R R

14 M-Lecture():.-. The solution is = -9, =, = - and = -9. amples ) Solve the sstemof linear equations 9 8 Solution

15 M-Lecture():.-. Solution: 7 9 linear equations the sstemof Solve ) w w w w

16 M-Lecture():.-. ) Solve the sstemof linear equations w 7w w Solution.

17 M-Lecture():.-. Note: () For the sstem of linear equations AX = B (B O), If X and X are two solutions, then rx + sx, r + s =, is also a solution. e.g. If X and X are two solutions to the sstem of linear equations AX = B (B O), then X X and.x +.7X e.g. If X, X and X are solutions to the sstem of linear equations AX = B (B O), then X + X X,is also a solution. () For the homogenous sstem of linear equations AX = O,If X and X are two solutions, then rx + sx is also a solution. e.g. If X and X are two solutions, then X + X and X X are also solutions. Note: The homogenous sstem is alwas consistent (has solution) which is either of following: The unique solution ( X = O, Zero solution), called the trivial solution, or an infinitel man solutions (including the trivial solution), called the nontrivial solution. 7

18 M-Lecture():.-. ample: Find all values of a for which thesstem (i) (ii) has no solution, has infinitel man solutions, (iii) has a unique solutions. ( a ) a Solution: ample: Let A. Solve the homogeneou s linear sstemax O. 8

19 M-Lecture():.-. 9 Solution: R t t t t t S t t t t O A,,,, Let man solutions. We have infinitel unknowns, than thenumber of less leading 's is Number of. Let Do book eercise.:,,,,7 9,,,,,7,8,,,,,,,7,8,,, 8,9,,, 9,