# 数学代写|线性代数代写linear algebra代考|MTH2106

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

Consider $A=\left[\begin{array}{ll}1 & 2 \ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \ 0 & 0\end{array}\right]$. Since it only takes one type-3 elementary row operation to get from $A$ to $B$, we see that $A \tilde{\mathrm{r}} B$ and that $\operatorname{rank} A=\operatorname{rank} B=$ 1. Notice, though, that $\operatorname{Col} A=\operatorname{Span}{(1,1)}$ while $\operatorname{Col} B=\operatorname{Span}{(1,0)}$. Row equivalent matrices have the same rank, but typically they have different column spaces. They do not, however, have different null spaces.

Lemma 4.17. Row equivalent matrices have identical null spaces.
Proof. Let $A$ belong to $M_{m, n}(\mathbb{F})$ and let $P$ the product of $m \times m$ elementary matrices over $\mathbb{F}$. Since $P$ is invertible, $A \mathbf{u}=\mathbf{0}$ if and only if
$$P A \mathbf{u}=P \mathbf{0}=\mathbf{0} .$$
This is enough to prove the lemma.
The null space of a matrix is the set of vectors with coordinates that list the coefficients in a dependence relation on the columns of the matrix. In particular, if $A=\left[\begin{array}{lll}\mathbf{a}1 & \cdots & \mathbf{a}_n\end{array}\right]$ belongs to $M{m, n}(\mathbb{F})$, then
$$u_1 \mathbf{a}_1+\cdots+u_n \mathbf{a}_n=\mathbf{0}_m$$
if and only if $\mathbf{u}=\left(u_1, \ldots, u_n\right)$ is in $\operatorname{Nul} A$.
This is what we mean by the following important theorem.
Theorem 4.18. Row equivalence preserves dependence relations on the columns of a matrix.

Example 4.19. Consider $A=\left[\begin{array}{rrrr}1 & 2 & -1 & 0 \ -1 & 1 & 2 & 1 \ -2 & 0 & 3 & -1\end{array}\right]$ over $\mathbb{R}$. The reduced row echelon form for $A$ is
$$B=\left[\begin{array}{rrrr} 1 & 0 & 0 & 11 \ 0 & 1 & 0 & -2 \ 0 & 0 & 1 & 7 \end{array}\right]$$
The following dependence relation on the columns of $B$ is immediately evident:
$$(11,-2,7)=11(1,0,0)-2(0,1,0)+7(0,0,1)$$

## 数学代写|线性代数代写linear algebra代考|An Early Use of the Determinant

Students learn to hand calculate determinants for $2 \times 2$ and $3 \times 3$ matrices in precalculus courses but may wonder what a determinant actually measures and how it came to be an object worthy of attention. In this section, we consider the work of Gabriel Cramer (1704-1752), who described and applied determinants to find a formula for the solution of a so-called nonsingular $n \times n$ system of linear equations over $\mathbb{R}$. There are other definitions of the determinant that we look at in later chapters and Cramer’s work was not the first known reference to determinants. It was an important influence, though, and it suggests some historical context for what can seem to be a puzzling object.

A determinant, in mathematics, is a number associated to a square matrix. A determinant, in English, is a deciding factor for something: The cost of feeding a dairy cow is a determinant in the price of butter. The determinant of an $n \times n$ system of linear equations is the deciding factor in whether the system has a uniquely determined – that is, well-defined – solution. A system with nonzero determinant has a solution set containing a single vector. Its solution is thus uniquely determined. A system with zero determinant has a solution set that is empty or that contains more than one vector. In either case, its solution is not uniquely determined.

Another curious technical word arises here. An $n \times n$ system of linear equations that has a unique solution may be called nonsingular. The coefficient matrix of a nonsingular system of linear equations is also said to be nonsingular. When applied to a matrix, nonsingular is a synonym for invertible. When applied to a system of linear equations, nonsingular means having a single solution. On the face of it, this may seem peculiar, less so if we dig a little deeper.

Singular means rare, or unexpected: The Big Bang was a singular event. Usage of the word nonsingular in the context of $n \times n$ systems of linear equations suggests that running across an $n \times n$ system of linear equations with no solutions, or more than one solution, would be a singular event. In some regard, this is not unreasonable.

Consider a $2 \times 2$ system of linear equations over $\mathbb{R}$. Each equation in the system represents a line in the $x y$-plane. The lines intersect in a unique point if and only if the system has exactly one solution. This is the nonsingular case. When the lines do not intersect, they are parallel, and the solution set is empty. When the two equations represent the same line $-x+y=1$ and $2 x+2 y=2$, for instance each point on the line is a solution, so the solution set is infinite. In the latter two settings, the lines have the same slope. This is the singular case.

# 线性代数代考

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

$\operatorname{Col} A=\operatorname{Span}(1,1)$ 尽管 $\operatorname{Col} B=\operatorname{Span}(1,0)$. 行等效矩阵具有相同的秩，但通常 它们具有不同的列空间。但是，它们没有不同的零空间。

$$P A \mathbf{u}=P \mathbf{0}=\mathbf{0} .$$

$$u_1 \mathbf{a}_1+\cdots+u_n \mathbf{a}_n=\mathbf{0}_m$$

$$(11,-2,7)=11(1,0,0)-2(0,1,0)+7(0,0,1)$$

## 数学代写|线性代数代写linear algebra代考|An Early Use of the Determinant

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