## 数学代写|线性代数代写linear algebra代考|MATH1002

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## 数学代写|线性代数代写linear algebra代考|THE SIMPLEX METHOD REVISITED

In this section we introduce a way of implementing the Simplex Method without the use of elementary row operations. Just as in Gaussian Elimination where we could replace elementary row operations by multiplication on the left by an elementary matrices, so too in the Simplex Method can we replace pivoting by multiplication on the left by an appropriate matrix in order to reach the next tableau.

First, it would make our discussion easier if we introduce matrix notation for representing a linear programming problem. Assume for this section that matrices with either one of their dimension being 1 are column matrices.

Let $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two matrices of the same dimensions. Then the notation $A, \leq$ and $\geq$ are defined in a sinilar manner.

Definition $2.13$ If our linear programming problem has $n$ unknowns and $m$ con traints, then the matrix representation of a standard linear programming problem has the form
Maximize $z=C^T X$ Subject to $A X \leq B, \quad X \geq 0$
where $A$ is an $m \times n$ matrix, $C$ is an $n \times 1$ matrix, $B$ is an $m \times 1$ matrix, and $X$ is an $n \times 1$ matrix of unknowns $x_1, x_2, \ldots, x_n$.

Definition 2.14 If our linear programming problem has $n$ unknowns and $m$ contraints, then the matrix representation of a canonical linear programming problem has the form Maximize $z=\left[C \mid 0_m\right]^T X^{\prime}$ Subject to $\left[A \mid I_m\right] X=B, \quad X \geq 0$
where $A$ is an $m \times n$ matrix, $C$ is an $n \times 1$ matrix, $B$ is an $m \times 1$ matrix, and $X^{\prime}$ is an $(m+n) \times 1$ matrix of unknowns $x_1, x_2, \ldots, x_n$ together with slack variables $u_1, u_2, \ldots, u_m$. Note that adding $m$ slack variables corresponds to replacing $A$ by $\left[A \mid I_m\right]$ and does nothing to effect the objective function, hence the $0_m$.

## 数学代写|线性代数代写linear algebra代考|NON-HOMOGENEOUS SYSTEMS AND RANK

In this section, we distinguish two kinds on linear systems of equations. There is a natural division of linear systems of equations into two types: homogeneous and non-homogeneous.

Definition 2.15 A homogeneous linear system of equations has the form $A X=0$, where 0 is a column of zeros. A non-homogeneous linear system of equations has the form $A X=B$, where $B \neq 0$, i.e. $B$ is a column with nonzero entries.
Example 2.33 The linear system below is homogeneous.
$$\left{\begin{array}{c} 2 x_1+x_2-x_3=0 \ x_1-3 x_2+x_3=0 \ -3 x_1+x_2+x_3=0 \end{array}\right.$$
The linear system below is non-homogeneous.
$$\left{\begin{array}{rlc} 2 x_1+x_2-x_3 & =-1 \ x_1-3 x_2+x_3 & = & 0 \ -3 x_1+x_2+x_3 & = & 6 \end{array}\right.$$
We remark that any homogeneous linear system has at least one solution, namely $X=0$ called the trivial solution. Some results can be proved at this point.
Theorem 2.14 For a square matrix A the following are equivalent:
i. A is invertible.
ii. A is equivalent to $I$.
iii. The linear system $A X=B$ has a unique solution for any $B \in \mathbb{R}^n$.
iv. A is a product of elementary matrices.
v. The linear system $A X=0$ has only the trivial solution.
Proof 2.16 Part iii certainly implies part v. To complete the proof, we show that part $v$ implies part ii. If $A X=0$ has only the trivial solution, then it must be the case that the augmented matrix $[A \mid 0]$ reduces to $[I \mid 0]$, and so $A$ is equivalent to $I$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|THE SIMPLEX METHOD REVISITED

$\left[A \mid I_m\right] X=B, \quad X \geq 0$

## 数学代写|线性代数代写linear algebra代考|NON-HOMOGENEOUS SYSTEMS AND RANK

$\$ \$$\sqrt{1} 左 2 x_1+x_2-x_3=0 x_1-3 x_2+x_3=0-3 x_1+x_2 \正确的。 Thelinearsystembelowisnon – homogeneous. 剩下{$$
2 x_1+x_2-x_3=-1 x_1-3 x_2+x_3=0-3 x
$$正确的。 \ \$$

i. $A$ 是可逆的。

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## MATLAB代写

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