# 线性代数代考_linear algebra代考_Introduction to Linear Algebra

## 线性代数代考_linear algebra代考_CALCULATION OF THE INVERSE WITH THE GAUSSIAN ALGORITHM

Suppose we have an invertible matrix $A$, and we want to compute its inverse. Consider the matrix:
$$M=(A \mid I)$$
obtained by putting the identity matrix next to $A$. Then, through elementary operations on the rows of the matrix $M$, we can get the identity matrix on the left-hand side. We briefly indicate the procedure and then we will clarify it with examples.
We already know how to get a matrix $C$ in row echelon form and, as the elementary row operations preserve the rank and the matrix $A$ is invertible, the matrix $C$ will have exactly $n$ pivots. Dividing each row by a suitable number, we can assume that all the pivots of $C$ are equal to 1 . We then proceed from “bottom” up, using the last pivot and appropriate elementary operations on rows to obtain a new matrix where the last pivot is 1 and in its column there are all zeros except for the last pivot itself. Then we move to the pivot before the last and perform the same procedure, until in the end the matrix on the left hand side of $M$ is the identity matrix. At this point, the matrix that appears to the right is the inverse matrix of $A$. We will not prove this procedure, but we will give examples.
Example 7.5.1 Consider the matrix
$$A=\left(\begin{array}{ll} 1 & 3 \ 1 & 4 \end{array}\right) .$$
To compute the inverse, we must apply the Gaussian algorithm to the matrix:
$$\left(\begin{array}{ll|ll} 1 & 3 & 1 & 0 \ 1 & 4 & 0 & 1 \end{array}\right) .$$
We carry out the following elementary operation: 2 nd row $\rightarrow$ nd row – 1st row, and we get:
$$\left(\begin{array}{cc|cc} 1 & 3 & 1 & 0 \ 0 & 1 & -1 & 1 \end{array}\right)$$

## 线性代数代考_linear algebra代考_THE LINEAR MAPS FROM

Now that we have introduced the concept of determinant and inverse of a matrix, we can give an important result that allows us to characterize invertible linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$.

Theorem 7.6.1 Let $F: \mathbb{R}^n \longrightarrow \mathbb{R}^n$ be a linear map, and let $A$ be the matrix associated to $F$ with respect to the canonical basis (in the domain and codomain). The following statements are equivalent.

1. $F$ is an isomorphism.
2. $F$ is injective.
3. $F$ is surjective.
4. $\operatorname{dim}(\operatorname{Im}(F))=n$.
5. $\operatorname{rk}(A)=n$.
6. The columns of $A$ are linearly independent.
7. The rows of $A$ are linearly independent.
8. The system $A \mathrm{x}=0$ has a unique solution.
9. For every $\mathbf{b} \subset \mathbb{R}^n$ the system $A \mathrm{x}=\mathrm{b}$ has a unique solution.
10. A is invertible.
11. The determinant of $A$ is not zero.
Proof. By Proposition 5.5.2, we immediately have the equivalence between (1), (2), (3). We now show that the statements (3) through (9) are equivalent, showing that each of them implies the next and then that (9) implies (2). We will show then, finally, that (1), (10), (11) are equivalent.
(3) implies (4), because if $F$ is surjective, then $\operatorname{Im} F=\mathbb{R}^n$ has dimension $n$.
(4) implies (5), because $\operatorname{rk}(A)=\operatorname{dim}(\operatorname{Im}(F))$, by Observation $6.2 .2$.
(5) implies (6), by the definition of rank of a matrix (which is in particular is the column rank).

# 线性代数代考

## 线性代数代考_linear algebra代考_CALCULATION OF THE INVERSE WITH THE GAUSSIAN ALGORITHM

$$M=(A \mid I)$$

$$A=\left(\begin{array}{llll} 1 & 3 & 1 & 4 \end{array}\right) .$$

$$\left(\begin{array}{ll|llllll} 1 & 3 & 1 & 0 & 1 & 4 & 0 & 1 \end{array}\right) .$$

$$\left(\begin{array}{cc|cccccc} 1 & 3 & 1 & 0 & 0 & 1 & -1 & 1 \end{array}\right)$$

## 线性代数代考_linear algebra代考_THE LINEAR MAPS FROM

1. $F$ 是一个同构。
2. $F$ 是单射的。
3. $F$ 是满射的。
4. $\operatorname{dim}(\operatorname{Im}(F))=n$.
5. $\operatorname{rk}(A)=n$.
6. 列的 $A$ 是线性独立的。
7. 的行 $A$ 是线性独立的。
8. 系统 $A \mathrm{x}=0$ 有唯一解。
9. 对于每一个 $\mathbf{b} \subset \mathbb{R}^n$ 系统 $A \mathbf{x}=\mathrm{b}$ 有唯一解。
10. $\mathrm{A}$ 是可逆的。
11. 的行列式 $A$ 不为零。
证明。根据命题 5.5.2，我们立即得到 (1)、(2)、(3) 之间的等价性。我们现在证明陈述 (3) 到 (9) 是等 价的，表明它们中的每一个都蕴含下一个，然后 (9) 蕴含 (2)。最后，我们将证明 (1)、(10)、(11) 是等 价的。
(3) 蕴含 (4)，因为如果 $F$ 是满射的，那么 $\operatorname{Im} F=\mathbb{R}^n$ 有维度 $n$.
(4) 蕴含 (5)， 因为 $r k(A)=\operatorname{dim}(\operatorname{Im}(F))$ ，通过观察 $6.2 .2$.
(5) 暗示 (6)，根据矩阵秩的定义 (特别是列秩)。

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