Arab Open University Faculty of Computer Studies M132: Linear Algebra
Course Contents Systems of Linear Equations Matrices and Matrix Operations Vectors, Linear Combinations and Linear Independence Vector Spaces, Subspaces, Span, Basis and Dimensions Linear Transformations, Null spaces and Ranges Eigenvalues and Eigenvectors
Systems of Linear Equations A set of equations is called a system of equations. A linear equation in n unknowns has the form where the variables are of first-degree. If all equations in a system are linear, the system is a system of linear equations, or a linear system. The solutions must satisfy each equation in the system.
Systems of Linear Equations Example Solve the linear system Solution (1) (2) Solve (2) for y Substitute y = x + 3 in (1) Solve for x Substitute x = 1 in y = x + 3 Solution set: {(1, 4)}
Systems of Linear Equations Matrices Information in science and mathematics is often organized into rows and columns to form rectangular arrays. Tables of numerical data that arise from physical observations 2×3
Linear Systems ~ Matrices Systems of Linear Equations Linear Systems ~ Matrices Example: (to solve linear equations) Solution is obtained by performing appropriate operations on this matrix
Systems of Linear Equations For any system of linear equations, we have 3 possibilities Unique solution Infinitely many solutions No solution y y y (2) (1) (2) (1) (1) (2) x x x (1,-1) x = 1, y = −1 Let y = t, x = 3 − t, t in R System is inconsistent
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations Definition: An m×n matrix A is said to be in reduced row echelon form if it satisfies the following conditions: All zero rows (consisting entirely of zeros), if any, are at the bottom. The first nonzero entry from the left of a nonzero row is 1, called the leading 1 for that row. Each leading 1 is to the right of all leading 1’s in the rows above it. Each leading 1 is to the only nonzero entry in its column. e.g.
Systems of Linear Equations
Systems of Linear Equations Definition: The elementary row operations on an m×n matrix A are: Interchanging two rows. Multiplying one row by a nonzero number. Add a multiple of one row to a different row. The matrix B is row equivalent to the matrices A.
Systems of Linear Equations Elementary Row Operations (Example) R2 = R2 – 2R1 R3 = R3 – 3R1
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations Let AX = B and CX = D be two systems of linear equations each of m equations in n unknowns. If the augmented matrices [A | B ] and [C | D ] of these systems are row equivalent, then both linear systems have exactly the same solutions. To solve the system AX = B : Form the augmented matrix [A | B ]. Find the matrix [C | D ] in reduced row echelon form that is row equivalent to the matrix [A | B ] that by using elementary row operations. For the matrix [C | D ], there are 3 possibilities: Number of leading 1’s = number of unknowns (variables), then the system has the unique solution X = D. Number of leading 1’s < number of unknowns, then the system has infinitely many solutions. Here the non-leading variables (unknowns corresponding to columns that do not contain leading 1) end up as parameters and the leading variables (unknowns corresponding to columns that contain leading 1) are given in terms of these parameters. The system is inconsistent (0 = 1 !!!), the system has no solution.
Systems of Linear Equations AX = B [A | B ] [C | D ] Unique solution Infinitely many solutions No solution
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations For the system of linear equations AX = B (B ≠ O), If X1 and X2 are two solutions, then rX1 + sX2 , r + s = 1, is also a solution. e.g. If X1 and X2 are two solutions to the system of linear equations AX = B (B ≠ O), then 3X1 - 2X2 and 0.25X1 + 0.75X2 are also solutions: e.g. If X1, X2 and X3 are solutions to the system of linear equations AX = B (B ≠ O), then 3X1 + 2X2 - 4X3 is also a solution.
Systems of Linear Equations For the homogenous system of linear equations AX = O, If X1 and X2 are two solutions, then rX1 + sX2 is also a solution. e.g. If X1 and X2 are two solutions, then 3X1 + 2X2 and 10X1 - 5X2 are also solutions. The homogenous system is always consistent (has solution) which is either of following: The unique solution ( X = O, Zero solution), called the trivial solution, or an infinitely many solutions (including the trivial solution), called the nontrivial solution.
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations