# 进一步线性代数|MATH0014 Algebra 3: Further Linear Algebra代写2023

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## Instructions:

Linear algebra is a fundamental subject in mathematics that has many applications in other fields such as physics, engineering, computer science, and economics.

In this course, you will deepen your understanding of linear algebra by exploring polynomial rings over fields, which are important in algebraic geometry and coding theory. You will also study matrix diagonalizability and the Jordan normal form, which are important for understanding the behavior of linear transformations and systems of linear equations.

In addition, you will learn about linear and bilinear forms, which are functions that take vectors as inputs and produce scalars as outputs. These concepts are important in the study of linear transformations and inner product spaces, which are vector spaces equipped with an additional structure that allows you to measure angles and distances between vectors.

Overall, this course will provide you with a deeper understanding of the key concepts of linear algebra and their applications, which will be useful in a wide range of fields.

Suppose $A$ is the matrix
$$A=\left[\begin{array}{llll} 0 & 1 & 2 & 2 \ 0 & 3 & 8 & 7 \ 0 & 0 & 4 & 2 \end{array}\right]$$
(a) Find all special solutions to $A x=0$ and describe in words the whole nullspace of $A$.

(a) Find all special solutions to $A x=0$ and describe in words the whole nullspace of $A$.
Solution: First, by row reduction
$$\left[\begin{array}{llll} 0 & 1 & 2 & 2 \ 0 & 3 & 8 & 7 \ 0 & 0 & 4 & 2 \end{array}\right] \rightarrow\left[\begin{array}{llll} 0 & 1 & 2 & 2 \ 0 & 0 & 2 & 1 \ 0 & 0 & 4 & 2 \end{array}\right] \rightarrow\left[\begin{array}{llll} 0 & 1 & 0 & 1 \ 0 & 0 & 2 & 1 \ 0 & 0 & 0 & 0 \end{array}\right] \rightarrow\left[\begin{array}{llll} 0 & 1 & 0 & 1 \ 0 & 0 & 1 & \frac{1}{2} \ 0 & 0 & 0 & 0 \end{array}\right]$$
so the special solutions are
$$s_1=\left[\begin{array}{l} 1 \ 0 \ 0 \ 0 \end{array}\right], s_2=\left[\begin{array}{c} 0 \ -1 \ -\frac{1}{2} \ 1 \end{array}\right]$$
Thus, $N(A)$ is a plane in $\mathbb{R}^4$ given by all linear combinations of the special solutions.

(b) Describe the column space of this particular matrix $A$. “All combinations of the four columns” is not a sufficient answer.

(b) Describe the column space of this particular matrix $A$. “All combinations of the four columns” is not a sufficient answer.

Solution: $C(A)$ is a plane in $\mathbb{R}^3$ given by all combinations of the pivot columns, namely
$$c_1\left[\begin{array}{l} 1 \ 3 \ 0 \end{array}\right]+c_2\left[\begin{array}{l} 2 \ 8 \ 4 \end{array}\right]$$

(c) What is the reduced row echelon form $R^*=\operatorname{rref}(B)$ when $B$ is the 6 by 8 block matrix
$$B=\left[\begin{array}{cc} A & A \ A & A \end{array}\right] \text { using the same } A \text { ? }$$

(c) What is the reduced row echelon form $R^*=\operatorname{rref}(B)$ when $B$ is the 6 by 8 block matrix
$$B=\left[\begin{array}{cc} A & A \ A & A \end{array}\right] \text { using the same } A \text { ? }$$
Solution: Note that $B$ immediately reduces to
$$B=\left[\begin{array}{cc} A & A \ 0 & 0 \end{array}\right]$$
We reduced $A$ above: the row reduced echelon form of of $B$ is thus
$$B=\left[\begin{array}{rr} \operatorname{rref}(A) & \operatorname{rref}(A) \ 0 & 0 \end{array}\right], \operatorname{rref}(A)=\left[\begin{array}{llll} 0 & 1 & 0 & 1 \ 0 & 0 & 1 & \frac{1}{2} \ 0 & 0 & 0 & 0 \end{array}\right]$$