## Problems ### 1 Vector Spaces #### 1.1 Definitions #### 1.2 Bases #### 1.3 Dimension of a Vector Space #### 1.4 Sums and Direct Sums ### 2 Matrices #### 2.1 The Space of Matrices #### 2.2 Linear Equations #### 2.3 Multiplication of Matrices ### 3 Linear Mappings #### 3.1 Mappings #### 3.2 Linear Mappings #### 3.3 The Kernel and Image of a Linear Map #### 3.4 Composition and Inverse of Linear Mappings #### 3.5 Geometric Applications ### 4 Linear Maps and Matrices #### 4.1 The Linear Map Associated with a Matrix #### 4.2 The Matrix Associated with a Linear Map #### 4.3 Bases, Matrices, and Linear Maps ### 5 Scalar Products and Orthogonality #### 5.1 Scalar Products #### 5.2 Orthogonal Bases, Positive Definite Case #### 5.3 Application to Linear Equations; the Rank #### 5.4 Bilinear Maps and Matrices #### 5.5 General Orthogonal Bases #### 5.6 The Dual Space and Scalar Products #### 5.7 Quadratic Forms #### 5.8 Sylvester's Theorem ### 6 Determinants #### 6.1 Determinants of Order 2 #### 6.2 Existence of Determinants #### 6.3 Additional Properties of Determinants #### 6.4 Cramer's Rule #### 6.5 Triangulation of a Matrix by Column Operations #### 6.6 Permutations #### 6.7 Expansion Formula and Uniqueness of Determinants #### 6.8 Inverse of a Matrix #### 6.9 The Rank of a Matrix and Subdeterminants ### 7 Symmetric, Hermitian, and Unitary Operators #### 7.1 Symmetric Operators #### 7.2 Hermitian Operators #### 7.3 Unitary Operators ### 8 Eigenvectors and Eigenvalues #### 8.1 Eigenvectors and Eigenvalues #### 8.2 The Characteristic Polynomial #### 8.3 Eigenvalues and Eigenvectors of Symmetric Matrices #### 8.4 Diagonalization of a Symmetric Linear Map #### 8.5 The Hermitian Case #### 8.6 Unitary Operators ### 9 Polynomials and Matrices #### 9.1 Polynomials #### 9.2 Polynomials of Matrices and Linear Maps ### 10 Triangulation of Matrices and Linear Maps #### 10.1 Existence of Triangulation #### 10.2 Theorem of Hamilton-Cayley #### 10.3 Diagonalization of Unitary Map ### 11 Polynomials and Primary Decomposition #### 11.1 The Euclidean Algorithm #### 11.2 Greatest Common Divisor #### 11.3 Unique Factorization #### 11.4 Application to the Decomposition of a Vector Space #### 11.S Schur's Lemma #### 11.6 The Jordan Normal Form ### 12 Convex Sets #### 12.1 Definitions #### 12.2 Separating Hyperplanes #### 12.3 Extreme Points and Supporting Hyperplanes #### 12.4 The Krein-Milman Theorem ### Appendix 1 Complex numbers 1. Express the following complex numbers in the form $x + iy$, where $x,$ $y$ are real numbers. 1. $(-1+3 i)^{-1}$ 2. $(1+i)(1-i)$ 3. $(1+i) i(2-i)$ 4. $(i-1)(2-i)$ 5. $(7+\pi i)(\pi+i)$ 6. $(2 i+1) \pi i$ 7. $(\sqrt{2}+i)(\pi+3 i)$ 8. $(i+1)(i-2)(i+3)$ 2. Express the following complex numbers in the form $x + iy$, where $x,$ $y$ are real numbers. 1. $(1+i)^{-1}$ 2. $\frac{1}{3+i}$ 3. $\frac{2+i}{2-i}$ 4. $\frac{1}{2-i}$ 5. $\frac{1+i}{i}$ 6. $\frac{i}{1+i}$ 7. $\frac{2 i}{3-i}$ 8. $\frac{1}{-1+i}$ 3. Let $\alpha$ be a complex number $\neq 0$. What is the absolute value of $\alpha / \bar{\alpha} ?$ What is $\bar{\alpha} ?$ 4. Let $\alpha, \beta$ be two complex numbers. Show that $\overline{\alpha \beta}=\bar{\alpha} \bar{\beta}$ and that $ \overline{\alpha+\beta}=\bar{\alpha}+\bar{\beta} $ 5. Show that $|\alpha \beta|=|\alpha||\beta|$ 6. Define addition of $n$-tuples of complex numbers componentwise, and multiplication of $n$-tuples of complex numbers by complex numbers componentwise also. If $A=\left(\alpha_{1}, \ldots, \alpha_{n}\right)$ and $B=\left(\beta_{1}, \ldots, \beta_{n}\right)$ are $n$-tuples of complex numbers, define their product $\langle A, B\rangle$ to be $\alpha_{1} \bar{\beta}_{1}+\cdots+\alpha_{n} \bar{\beta}_{n}$ (note the complex conjugation!). Prove the following rules: - HP 1. $\langle A, B\rangle=\overline{\langle B, A\rangle}$. - HP 2. $\langle A, B+C\rangle=\langle A, B\rangle+\langle A, C\rangle$. - HP 3. If $\alpha$ is a complex number, then$\langle\alpha A, B\rangle=\alpha\langle A, B\rangle \quad \text { and }\langle A, \alpha B\rangle=\bar{\alpha}\langle A, B\rangle$ - HP 4. If $A=O$ then $\langle A, A\rangle=0$, and otherwise $\langle A, A\rangle>0$ 7. We assume that you know about the functions sine and cosine, and their addition formulas. Let $\theta$ be a real number. 1. Define $e^{i \theta}=\cos \theta+i \sin \theta$ 2. Show that if $\theta_{1}$ and $\theta_{2}$ are real numbers, then$e^{i\left(\theta_{1}+\theta_{2}\right)}=e^{i \theta_{1}} e^{i \theta_{2}}$ Show that any complex number of absolute value 1 can be written in the form $e^{i t}$ for some real number $t$. 3. Show that any complex number can be written in the form $r e^{i \theta}$ for some real numbers $r, \theta$ with $r \geqq 0$. 4. If $z_{1}=r_{1} e^{i \theta_{1}}$ and $z_{2}=r_{2} e^{i \theta_{2}}$ with real $r_{1}, r_{2} \geqq 0$ and real $\theta_{1}, \theta_{2}$, show that $z_{1} z_{2}=r_{1} r_{2} e^{i\left(\theta_{1}+\theta_{2}\right)}$ 5. If $z$ is a complex number, and $n$ an integer gt;0$, show that there exists a complex number $w$ such that $w^{n}=z$. If $z \neq 0$ show that there exists $n$ distinct such complex numbers $w$. (Hint: If $z=r e^{i \theta}$, consider first $r^{1 / n} e^{i \theta / n}$.) 8. Assuming the complex numbers algebraically closed, prove that every irreducible polynomial over the real numbers has degree 1 or 2 . (Hint: Split the polynomial over the complex numbers and pair off complex conjugate roots.) ### Appendix 2 Iwasawa Decomposition and Others