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• Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$ Let $T:\R^2 \to \R^3$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 4\\ 3 \end{bmatrix} […]
• Trace of the Inverse Matrix of a Finite Order Matrix Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix. Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that […]
• Hyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors \[\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n\] satisfying the linear equation of the form \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]
• Normal Subgroups, Isomorphic Quotients, But Not Isomorphic Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$. Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.   Proof. We give a […]
• Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]
• 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 […]
• If the Quotient Ring is a Field, then the Ideal is Maximal Let $R$ be a ring with unit $1\neq 0$. Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$. (Do not assume that the ring $R$ is commutative.)   Proof. Let $I$ be an ideal of $R$ such that \[M \subset I \subset […]
• Positive definite Real Symmetric Matrix and its Eigenvalues A real symmetric $n \times n$ matrix $A$ is called positive definite if \[\mathbf{x}^{\trans}A\mathbf{x}>0\] for all nonzero vectors $\mathbf{x}$ in $\R^n$. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive. (b) Prove that if […]