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# 数学代写|数值分析代写Numerical analysis代考|MATH2722 Review of Matrix Algebra

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## 数学代写数值分析代写Numerical analysis代考|Matrices and Vectors

A matrix is a rectangular array of numbers. Thus,
$$\left(\begin{array}{rrr} 1 & 0 & 3 \ -2 & 4 & 1 \end{array}\right), \quad\left(\begin{array}{cc} \pi & 0 \ e & \frac{1}{2} \ -1 & .83 \ \sqrt{5} & -\frac{4}{7} \end{array}\right), \quad\left(\begin{array}{lll} .2 & -1.6 & .32 \end{array}\right), \quad\left(\begin{array}{c} 0 \ 0 \end{array}\right), \quad\left(\begin{array}{rr} 1 & 3 \ -2 & 5 \end{array}\right),$$
are all examples of matrices. We use the notation
$$A=\left(\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 n} \ \vdots & \vdots & \ddots & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right)$$
for a general matrix of size $m \times n$ (read ” $m$ by $n$ “), where $m$ denotes the number of rows in $A$ and $n$ denotes the number of columns. Thus, the preceding examples of matrices have respective sizes $2 \times 3,4 \times 2,1 \times 3,2 \times 1$, and $2 \times 2$. A matrix is square if $m=n$, i.e., it has the same number of rows as columns. A column vector is a $m \times 1$ matrix, while a row vector is a $1 \times n$ matrix. As we shall see, column vectors are by far the more important of the two, and the term “vector” without qualification will always mean “column vector”. A $1 \times 1$ matrix, which has but a single entry, is both a row and a column vector.

The number that lies in the $i^{\text {th }}$ row and the $j^{\text {th }}$ column of $A$ is called the $(i, j)$ entry of $A$, and is denoted by $a_{i j}$. The row index always appears first and the column index second. Two matrices are equal, $A=B$, if and only if they have the same size, and all their entries are the same: $a_{i j}=b_{i j}$ for $i=1, \ldots, m$ and $j=1, \ldots, n$.

## 数学代写|数值分析代写Numerical analysis代考|Matrices and Vectors

A general linear system of $m$ equations in $n$ unknowns will take the form
$$\begin{array}{cc} a_{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n=b_1, \ a_{21} x_1+a_{22} x_2+\cdots+a_{2 n} x_n=b_2, \ \vdots & \vdots \ a_{m 1} x_1+a_{m 2} x_2+\cdots+a_{m n} x_n=b_m \end{array}$$
As such, it is composed of three basic ingredients: the $m \times n$ coefficient matrix $A$, with entries $a_{i j}$ as in (3.1), the column vector $\mathbf{x}=\left(\begin{array}{c}x_1 \ x_2 \ \vdots \ x_n\end{array}\right)$ containing the unknowns, and the column vector $\mathbf{b}=\left(\begin{array}{c}b_1 \ b_2 \ \vdots \ b_m\end{array}\right)$ containing right hand sides. As an example, consider the linear system
$$\begin{array}{r} x+2 y+z=2, \ 2 y+z=7, \ x+y+4 z=3, \end{array}$$
The coefficient matrix $A=\left(\begin{array}{lll}1 & 2 & 1 \ 0 & 2 & 1 \ 1 & 1 & 4\end{array}\right)$ can be filled in, entry by entry, from the coefficients of the variables appearing in the equations. (Don’t forget to put a zero when a avariable doesn’t appear in an equation!) The vector $\mathbf{x}=\left(\begin{array}{l}x \ y \ z\end{array}\right)$ lists the variables, while the entries of $\mathbf{b}=\left(\begin{array}{l}2 \ 7 \ 3\end{array}\right)$ are the right hand sides of the equations.

Remark: We will consistently use bold face lower case letters to denote vectors, and ordinary capital letters to denote general matrices.
Matrix Arithmetic
Matrix arithmetic involves three basic operations: matrix addition, scalar multiplication, and matrix multiplication. First we define addition of matrices. You are only allowed to add two matrices of the same size, and matrix addition is performed entry by entry. For example,
$$\left(\begin{array}{rr} 1 & 2 \ -1 & 0 \end{array}\right)+\left(\begin{array}{rr} 3 & -5 \ 2 & 1 \end{array}\right)=\left(\begin{array}{rr} 4 & -3 \ 1 & 1 \end{array}\right)$$

## 数学代写|数值分析代写Numerical analysis代考|Matrices and Vectors

$$a_{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n=b_1, a_{21} x_1+a_{22} x_2+\cdots+a_{2 n} x_n=b_2, \vdots \quad \vdots a_{m 1} x_1+a_{m 2} x_2+\cdots+a_{m n} x_n=b_m$$

$$x+2 y+z=2,2 y+z=7, x+y+4 z=3,$$

## MATLAB代写

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