Systems Of Linear Equations Assignment Help
Introduction to the System of linear equation
A linear equation in variables x1, x2, x3, xn is an equation of form a1 x1+ a2 x2+........... + anxn =b
Where a1, a2, a3................an and b are constant real or complex number. A system of linear equations is a finite collection of linear equations involving the same set of variables.
General form for system of linear equations:
Vector equation: One extremely helpful view is that each unknown is a weight for a column vector in a linear combination.
Matrix equation: The vector equation is equivalent to a matrix equation of the form
Ax = b
Where A is an mn matrix, x is a column vector with n entries and b is a column vector with m entries.
The number of vectors in a basis for the span is now expressed as the rank of the matrix.
Properties of System of linear Equation:
The four expressions of a linear system:
1.A general system of linear equations can be written a
a11x1 + a12x2 + ...... + a1nxn = b1
a21x1 + a22x2 + ......... + a2nxn = b1
am1x1 + am2x2 + ...... + amnxn = bm
The vector equation form: x1a1 + x2a2 + + xnan = b;
3. The matrix equation form: Ax = b
4. The augmented matrix form: [a1, a2......., an | b]
Several algorithms for solving a linear equation:
- Describing the solution
- Elimination of variables
- Row reduction
- Cramer's rule
- Matrix solution
- Other methods
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