# 数学代写|线性代数代写linear algebra代考|MATH1051

## 数学代写|线性代数代写linear algebra代考|LU-Factorization

LU-factorization is a technique that can be applied to solving systems of linear equations, inverting matrices, and calculating determinants. In this section, we consider the ideas that support LU-factorization and we look at a few low-rank examples.

The optimal setting for solving a system of linear equations is one in which we can get the solution either by back-substitution or by forward-substitution. While this does not make an enormous difference for small systems of equations that we solve by hand, the computational savings this setting affords is magnified considerably in large systems of equations solved using machines. This is the driving idea behind LU-factorization. LU-factorization can be applied anytime we have a coefficient matrix that can be put into row echelon form using only downward pivoting. Some terminology helps facilitate our discussion.

A matrix $A=\left[a_{i j}\right]$ in $M_n(\mathbb{F})$ is lower triangular provided $a_{i j}=0$ for $ij, A$ is upper triangular. The matrix $A$ is a diagonal matrix if it is both upper and lower triangular: $a_{i j}=0$ if $i \neq j$.

The following matrices are, respectively, lower triangular, upper triangular, and diagonal:
$$\left[\begin{array}{lll} 1 & 0 & 0 \ 2 & 4 & 0 \ 1 & 2 & 3 \end{array}\right], \quad\left[\begin{array}{lll} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 3 \end{array}\right], \quad\left[\begin{array}{lll} 2 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 1 \end{array}\right]$$

When the coefficient matrix for a system of linear equations is upper triangular, we can solve the system using back-substitution. When the coefficient matrix for a system of linear equations is lower triangular, we can solve the system using forward-substitution.

Example 4.33. Let $A=\left[\begin{array}{lll}1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 3\end{array}\right]$ and let $\mathbf{b}=(-1,1,2)$. We solve $A \mathbf{x}=\mathbf{b}$ for $\mathbf{x}=(x, y, z)$. The last equation in the system of linear equations associated to $A \mathbf{x}=\mathbf{b}$ is $3 z=2$ so $z=2 / 3$. Back-substituting into $y+4 z=1$, we get $y=-5 / 3$. Back-substituting into the first equation, we get
$$x+2 y+3 z=x+(2)(-5 / 3)+(3)(2 / 3)=-1$$
so $x=1 / 3$.
A square matrix in row echelon form is upper triangular. An arbitrary matrix $A=\left[a_{i j}\right]$ in row echelon form may not be upper triangular, but it still has the property that $a_{i j}=0$ when $i>j$ and it is still the case that solving an associated system of linear equations only requires back-substitution.

Upper and lower triangular matrices have nice mathematical properties, as well.

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

We have already discussed the vector space of linear transformations from one vector space to another. Perhaps counterintuitively, the mappings from a vector space into its scalar field are particularly interesting. They are called linear functionals.
Example 5.1. Let $V=C([0,1])$ where $[0,1] \subseteq \mathbb{R}$. We leave it as an exercise to verify that
$$f \mapsto \int_0^1 f(t) d t$$
defines a linear functional on $V$.
The set of all linear functionals on a vector space is its dual space. We denote the dual space of $V$ by $V^$. Since $V^=\mathcal{L}(V, \mathbb{F})$, where $\mathbb{F}$ is the scalar field for $V$, Corollary $3.28$ tells us that if $\operatorname{dim} V=n$, then $\operatorname{dim} V^*=n$, as well.

When $V$ is infinite-dimensional, it may not be the case that $V \cong V^$. In that setting, $V^$ may be called the algebraic dual or the full algebraic dual of $V$ and “dual space” may refer to a proper subspace of $V^*$.

Since a linear mapping $\mathbb{F}^n \rightarrow \mathbb{F}$ is effected by premultiplication by an $n \times 1$ matrix, the following is immediate.
Theorem 5.2. The dual space of $\mathbb{F}^n$ is $\mathbb{F}_n$.
This is the interesting relationship between $\mathbb{F}^n$ and $\mathbb{F}_n$ that we promised in Section 1.2. It also suggests a deeper relationship between duality and matrix

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|LU-Factorization

LU 分解是一种可用于求解线性方程组、求逆矩阵和计算行列式的技术。在本节中， 我们考虑支持 LU 分解的想法，并查看一些低秩示例。

$$x+2 y+3 z=x+(2)(-5 / 3)+(3)(2 / 3)=-1$$

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

$$f \mapsto \int_0^1 f(t) d t$$

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