# Nagoya-university-exam-eye-catch

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• Determine Trigonometric Functions with Given Conditions (a) Find a function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants. (b) Find real numbers $a, b, c$ such that the function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […] • Orthonormal Basis of Null Space and Row Space Let A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}. (a) Find an orthonormal basis of the null space of A. (b) Find the rank of A. (c) Find an orthonormal basis of the row space of A. (The Ohio State University, Linear Algebra Exam […] • Finite Order Matrix and its Trace Let A be an n\times n matrix and suppose that A^r=I_n for some positive integer r. Then show that (a) |\tr(A)|\leq n. (b) If |\tr(A)|=n, then A=\zeta I_n for an r-th root of unity \zeta. (c) \tr(A)=n if and only if A=I_n. Proof. (a) […] • Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let A be a 3\times 3 matrix. Suppose that A has eigenvalues 2 and -1, and suppose that \mathbf{u} and \mathbf{v} are eigenvectors corresponding to 2 and -1, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […] • Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations. (a) \left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]. (b) […] • The Ring \Z[\sqrt{2}] is a Euclidean Domain Prove that the ring of integers \[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}$ of the field $\Q(\sqrt{2})$ is a Euclidean Domain.   Proof. First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$. We use the […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […] • Trace, Determinant, and Eigenvalue (Harvard University Exam Problem) (a) A$2 \times 2$matrix$A$satisfies$\tr(A^2)=5$and$\tr(A)=3$. Find$\det(A)$. (b) A$2 \times 2$matrix has two parallel columns and$\tr(A)=5$. Find$\tr(A^2)$. (c) A$2\times 2$matrix$A$has$\det(A)=5\$ and positive integer eigenvalues. What is the trace of […]