# MA 405Introduction to Linear AlgebraSection: 601

## Course Description

This course offers a rigorous treatment of linear algebra, including systems of linear equations, matrices, determinants, abstract vector spaces, bases, linear independence, spanning sets, linear transformations, eigenvalues and eigenvectors, similarity, inner product spaces, orthogonality and orthogonal bases, factorization of matrices. Compared with MA 305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. MA 225 is recommended as a prerequisite. Credit is not allowed for both MA 305 and MA 405

### DE Program

Mathematics-CTG

## 2210Q 应用线性代数Applied Linear Algebra题目解答

A matrix $A \in M_{2}(\mathbf{C})$ is invertible if and only if $\operatorname{det} A \neq 0$. If this is the case, then $$ A^{-1}=\frac{1}{\operatorname{det} A}\left(\operatorname{Tr}(A) \cdot I_{2}-A\right) $$

Suppose that $A$ is invertible. Then taking the determinant in the equality $A \cdot A^{-1}=I_{2}$ we obtain $$ \operatorname{det} A \cdot \operatorname{det} A^{-1}=\operatorname{det} I_{2}=1 $$ thus $\operatorname{det} A \neq 0$. Conversely, suppose that $\operatorname{det} A \neq 0$ and define $$ B=\frac{1}{\operatorname{det} A}\left(\operatorname{Tr}(A) \cdot I_{2}-A\right) $$ Then using the Cayley-Hamilton theorem we obtain $$ A B=\frac{1}{\operatorname{det} A}\left(\operatorname{Tr}(A) \cdot A-A^{2}\right)=\frac{1}{\operatorname{det} A} \cdot \operatorname{det} A I_{2}=I_{2} $$ and similarly $B A=I_{2}$. Thus $A$ is invertible and $A^{-1}=B$.

Let $A, B \in M_{2}(\mathbf{C})$ be two matrices such that $A B=I_{2}$. Then $A$ is invertible and $B=A^{-1}$. In particular, we have $B A=I_{2}$.

Since $A B=I_{2}$, we have $\operatorname{det} A \cdot \operatorname{det} B=\operatorname{det}(A B)=1$, thus $\operatorname{det} A \neq 0$. The previous theorem shows that $A$ is invertible. Multiplying the equality $A B=I_{2}$ by $A^{-1}$ on the left, we obtain $B=A^{-1}$. Finally, $B A=A^{-1} A=I_{2}$.

By the previous theorem, we can find two nonzero vectors $X_{1}=\left[\begin{array}{l}x_{11} \ x_{21}\end{array}\right]$ and $X_{2}=\left[\begin{array}{l}x_{12} \ x_{22}\end{array}\right]$ such that $A X_{i}=\lambda_{i} X_{i}$.

Consider the matrix $P=\left[\begin{array}{ll}x_{11} & x_{12} \ x_{21} & x_{22}\end{array}\right]$ whose columns are $X_{1}, X_{2}$. A simple computation shows that the columns of $A P$ are $\lambda_{1} X_{1}$ and $\lambda_{2} X_{2}$, which are the columns of $P\left[\begin{array}{cc}\lambda_{1} & 0 \ 0 & \lambda_{2}\end{array}\right]$, thus $A P=P\left[\begin{array}{cc}\lambda_{1} & 0 \ 0 & \lambda_{2}\end{array}\right]$. It remains to see that if $\lambda_{1} \neq \lambda_{2}$, then $P$ is invertible (we haven’t used so far the hypothesis $\lambda_{1} \neq \lambda_{2}$ ).

Suppose that det $P=0$, thus $x_{11} x_{22}=x_{21} x_{12}$. This easily implies that the columns of $P$ are proportional, say the second column $X_{2}$ is $\alpha$ times the first column, $X_{1}$. Thus $X_{2}=\alpha X_{1}$. Then $$ \lambda_{2} X_{2}=A X_{2}=\alpha A X_{1}=\alpha \lambda_{1} X_{1}=\lambda_{1} X_{2}, $$ forcing $\left(\lambda_{1}-\lambda_{2}\right) X_{2}=0$. This is impossible as both $\lambda_{1}-\lambda_{2}$ and $X_{2}$ are nonzero. The problem is solved.

Solve in $M_{2}(\mathbf{C})$ the following equations (a) $A^{2}=O_{2}$. (b) $A^{2}=I_{2}$. (c) $A^{2}=A$.

(a) Let $A$ be a solution of the problem. Then $\operatorname{det} A=0$ and the CayleyHamilton relation reduces to $\operatorname{Tr}(A) A=0$. Taking the trace yields $\operatorname{Tr}(A)^{2}=0$, thus $\operatorname{Tr}(A)=0$. Conversely, if $\operatorname{det} A=0$ and $\operatorname{Tr}(A)=0$, then the CayleyHamilton theorem shows that $A^{2}=O_{2}$. Thus the solutions of the problem are the matrices $A=\left[\begin{array}{cc}a & b \ c & -a\end{array}\right], \quad$ with $\quad a, b, c \in \mathbf{C} \quad$ and $\quad a^{2}+b c=0 .$ (b) We must have det $A=\pm 1$ and, by the Cayley-Hamilton theorem, $I_{2}-$ $\operatorname{Tr}(A) A+\operatorname{det} A I_{2}=O_{2}$. If det $A=1$, then $\operatorname{Tr}(A) A=2 I_{2}$ and taking the trace yields $\operatorname{Tr}(A)^{2}=4$, thus $\operatorname{Tr}(A)=\pm 2$. This yields two solutions, $A=\pm I_{2}$. Suppose that $\operatorname{det} A=-1$. Then $\operatorname{Tr}(A) A=O_{2}$ and taking the trace gives $\operatorname{Tr}(A)=0$. Conversely, any matrix $A$ with $\operatorname{Tr}(A)=0$ and $\operatorname{det} A=-1$ is a solution of the problem (again by Cayley-Hamilton). Thus the solutions of the equation are $\pm I_{2} \quad$ and $\quad A=\left[\begin{array}{cc}a & b \ c & -a\end{array}\right], \quad a, b, c \in \mathbf{C}, \quad a^{2}+b c=1 .$ (c) If $\operatorname{det} A \neq 0$, then multiplying by $A^{-1}$ yields $A=I_{2}$. So suppose that $\operatorname{det} A=$ 0 . The Cayley-Hamilton theorem yields $A-\operatorname{Tr}(A) A=O_{2}$. If $\operatorname{Tr}(A) \neq 1$, this forces $A=O_{2}$, which is a solution of the problem. Thus if $A \neq O_{2}, I_{2}$, then det $A=0$ and $\operatorname{Tr}(A)=1$. Conversely, all such matrices are solutions of the problem (again by Cayley-Hamilton). Thus the solutions of the problem are $O_{2}, \quad I_{2} \quad$ and $\quad A=\left[\begin{array}{cc}a & b \ c & 1-a\end{array}\right], \quad a, b, c \in \mathbf{C}, \quad a^{2}+b c=a .$

## 线性代数和矩阵导论MA 405. Introduction to Linear Algebra and Matrices

Instructor: V. Summers | SAS 4121 | vwsummer@ncsu.edu | Moodle page: http://moodle.wolfware.ncsu.edu Lectures will be held MWF 9:35 am – 10:25 am in SAS 2235. Communication: Office hours will be held Mondays 10:30 – 12:30 in SAS 4121. Moodle forums will be used for most communications in this course. You are encouraged to discuss concepts and homework in the forums with your fellow students; the discussions will be monitored by Mr. Summers. Course text There is no assigned textbook for this course. The class notes and video lectures are available in Moodle. The following texts are good resources:

- Linear Algebra Done Right, by Sheldon Axler, Springer International Publishing : Imprint: Springer, 2015 ISBN: 9783319110806 – available through NCSU libraries.
- Linear Algebra – A Free text for a standard US undergraduate course, by Jim Hefferon – link to open source book available in Moodle The Math Multimedia Center is a tutorial center for undergraduate students that need help in their mathematics courses (100- through 300-level), and is staffed by math graduate students familiar with the material taught in these courses. Location: SAS Hall 2103/2105 Hours: Monday – Friday 8:00 am – 5:00 pm You can also get help with your courses (not only math) at the NCSU Undergraduate Tutorial Center. Catalog Description Prerequisite: MA 241 (Co-requisite MA 242) This course offers a rigorous treatment of linear algebra, including systems of linear equations, matrices, determinants, abstract vector spaces, bases, linear independence, spanning sets, linear transformations, eigenvalues and eigenvectors, similarity, inner product spaces, orthogonality and orthogonal bases, factorization of matrices. Compared with MA 305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. MA 225 is recommended as a prerequisite. Credit is not allowed for both MA 305 and MA 405. Course overview Linear Algebra provides one of the cornerstones for much of modern Mathematics, and has important applications in Physics, Chemistry, Engineering, Economics, Game Theory, Cryptography, Differential Equations, Genetics, Coding Theory and Sociology, and just about any other field of study involving mathematics you can imagine. The main purpose of this course is to introduce the basic concepts from linear algebra, explain the underlying theory, the computational techniques, and study how these concepts and results can be productively used in other areas of mathematics and physical sciences. Among the topics covered in this course will be: solving systems of linear equations using Gauss elimination, row echelon form, determinants, vector spaces, linear independence, bases, dimension, linear maps, orthogonality, eigenvalues, and reduction of matrices to diagonal forms. If time permits, we will discuss applications of linear algebra to differential equations and/or Fibonacci sequences. The subject involves a mixture of both the practical and the theoretical, and will provide in particular a good introduction to mathematical proofs. The student should be prepared to invest considerable amount of time in understanding the class material and doing homework. Credit is not allowed for both MA 305 and MA 405. Learning Objectives Upon successful completion of this course, students will be able to:
- Use Mathematical Notation and Terminology. The students will demonstrate mastery in using the mathematical notation and terminology of linear algebra. Students will read, interpret, and use the vocabulary, symbolism and basic definitions.
- Understand and Describe the Fundamental Concepts of Linear Algebra. Students will identify and apply the theorems about abstract vector spaces and linear transformations; will gain a clear understanding of the basic concepts of linear algebra, such as linear independence of vectors, spanning sets, basis, dimension, NCSU Department of Mathematics Spring 2019 linear maps, isomorphism, similarity, eigenvalues and eigenvectors.
- Identify and Utilize Linear Algebra Tools. The students will be able to apply course material along with techniques and procedures covered in this course to solve problems. Students will master techniques for solving linear systems by various matrix methods, compute the determinant and the inverse of a square matrix, work with matrix representations of linear maps, compute various factorizations of matrices, apply the Gram-Schmidt process, calculate and analyze the characteristic equation of a matrix to determine its eigenvalues and eigenvectors. Moreover, students will apply properties and theorems about vector spaces to specific mathematical structures that satisfy the vector space axioms, will analyze the differences and similarities between spanning sets, bases, and orthogonal bases and will use the knowledge gained in this course to determine appropriate methods of proof for specific problems.
- Develop Cognitive Skills. Students will demonstrate the ability to reason with abstract linear algebra concepts, to read and comprehend mathematical arguments utilizing direct and indirect proof, case analysis, and mathematical induction. Students will develop familiarity with axiomatic approach in mathematics through the study of vector spaces and linear transformations. They will acquire a level of proficiency in manipulating linear algebra concepts, in analyzing and evaluating their applicability in their future studies, including graduate work, in academic areas requiring linear algebra as a prerequisite for work in occupational fields requiring a background in linear algebra.

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