System of linear equations
Definition
\[ \begin{aligned} a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n} &=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n} &=b_{2} \\ \vdots \\ a_{m 1} x_{1}+a_{m 2} x_{2}+\cdots+a_{m n} x_{n} &=b_{m} \end{aligned} \]
\[ A=\left[\begin{array}{cccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\cdots} & {a_{2 n}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {a_{m 1}} & {a_{m 2}} & {\cdots} & {a_{m n}}\end{array}\right], \vec{b}=\left[b_{1} b_{2} \cdots b_{m}\right], \vec{x}=\left[\begin{array}{ccc}{x_{1}} & {x_{2}} & {\cdots} & {x_{n}}\end{array}\right] \]
\ (A: \) Coefficient the Matrix (coefficient matrix), \ (\ VEC B \) : the Constants (constant), \ (\ VEC X \) : the Variables
Example
\[ .80x_1 + .60x_2 + .40x_3 = 5\\ .20x_1 + .40x_2 + .60x_3 = 3 \]
Solution Set
Definition
The set of all solutions of a system of linear equations is called Solution Set
\[ solution~set = \lbrace \left[\begin{array}{cc}x_1\\x_2\\\vdots\\ x_n \end{array}\right]\in \mathcal{R}^n|x_1, x_2, \cdots, x_n~satisfy~the~system~of~linear~equations \rbrace \]
Example
Linear TWO Variables in Equation A and Y has X The form \ (AX by + C = \) , When A and AT B Least One of nonzero IS, IS The Equation of the this in The Line A \ (XY \) Plane as , THUS a System of linear equations 2 the Variables X and Y in Consists of equations a pair of, in each of Line a DESCRIBE the plane. (two linear equations solution may be understood as a straight line on a two-dimensional plane intersection case)
\ [A_1 = X + Y B_1 c_1 and \ qquad \ {IS The text of Line Equation $ \ mathcal L_1} {} \\ $ A_2 = X + Y B_2 c_2 \ qquad \ {IS The text of Equation $ Line \ mathcal L_2 are {}} $ \]
only One Solution (two lines intersect)
Solutions MANY infinitely (two coincident straight line)
Solution NO (two parallel straight lines do not overlap)
Statement
Every system of linear equations has no solution, exactly one solution, or infinitely many solutions.
A system of linear equations that has one or more solutions is called consistent; Otherwise, the system is inconsistent.
Two system of linear equations are called equivalent if they have exactly the same solution set.
Elementary Row Operations
Definition
Interchage any two rows of the matrix. (Interchange operation)
Multiply every entry of some row of the matrix by the same nonzero scalar. (scaling operation)
Add a multiple of one row of the matrix to another row. (row addition operation)
\ (\ mathbf X = {A} B \) , $ [\ mathbf {A} ~ B] $ IS Called The Augmented Matrix (augmented matrix)
Property
Every elementary row operations are reversble (elementary row operations are reversible)
\[ \mathbf{A}x = b \Longleftrightarrow \mathbf{A'}x = b' \quad \text{(equivalent)}\\ [\mathbf{A}~b] \longleftrightarrow [\mathbf{A'~}b'] \quad \text{(elmentary row opeations)} \]
Elementary row operations taken on the system of linear equations as well as on the corresponding augmented matrices. (Elementary row operations on a linear set of equations is equivalent to perform related operations on the augmented matrix)
Row echelon form
Definition
- Each nonzero row lies above every zero row. (Above a certain non-zero row of all zeros in rows)
- The leading entry of a nonzero row lies in a column to the right of the column containing the leading entry of any preceding row.
- If a column contains the leading entry of some row, the all entries of that column below the leading entry are 0. (If the first item of a column contains a row, then all of the first item of the following elements are 0)
if a matrix also satisfied the following two additional conditions, we say that it is in reduced echelon form.
- The column A leading the contains IF entry of some Row, The All Others The entries It of that column are of 0. The
(first item if the column contains a row, then the addition to this all elements of the first item out of the column is 0) - The leading entry of each nonzero row is (the first item are all non-zero rows 1) 1
Example
Consider leading entry row, each row may be leading entry becomes basic variables, diverted to other entry can be called free variables, free variables could be any value, but not necessarily of linear equations which are present in each system free variable , consider equations
\ [\ begin {aligned} x1 - 3x_2 ~~~~~~~ + 2x_4 ~~~~~~ = 7 \\ x_3 + 6x_4 ~~~~~~ = 9 \\ x_5 = 0 = 0 \\ \\ 2 \ the aligned End {} \]
of The Equations the Resulting
\ [\ the aligned the begin {} & 3x_2 x_1 = - \\ & 2x_4. 7 + x_2 = \ Free text {variable} = \\ & X_3. 9 - 6x_4 \ \ & x_4 = \ text {free
variable} \\ & x_5 = 2 \ end {aligned} \] can be obtained through solution of equations (General solution)
\ [\} x_1 bmatrix the begin {\\ x_2 X_4 \\ \\ \\ X_3 x_5 \ end {bmatrix} = \ begin {bmatrix} 7 \\ 0 \\ 9 \\ 0 \\ 2 \ end {bmatrix} + x_2 \ begin {bmatrix} 3 \\ 1 \\ 0 \\ 0 \\ 0 \ end {bmatrix} + x_4 \ begin {bmatrix} -2 \\ 0 \\ -6 \\ 1 \\ 0 \ end {bmatrix} \]
Summary
General Procedure for solving linear equations is the augmented matrix (augmented matrix) into its most simple row stepped \ ([\ mathbf {A} ~ b] \ xrightarrow {} [\ mathbf {R} ~ c] \ )
Then to seek basic variable by-pass and solution free variable