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  • Determine a 2-Digit Number Satisfying Two ConditionsDetermine a 2-Digit Number Satisfying Two Conditions A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45. Find the number.   Solution. The key to this problem is noticing that our 2-digit number can be […]
  • A Linear Transformation Maps the Zero Vector to the Zero VectorA Linear Transformation Maps the Zero Vector to the Zero Vector Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively. Show that $T(\mathbf{0}_n)=\mathbf{0}_m$. (The Ohio State University Linear Algebra […]
  • Show that Two Fields are Equal: $\Q(\sqrt{2}, \sqrt{3})= \Q(\sqrt{2}+\sqrt{3})$Show that Two Fields are Equal: $\Q(\sqrt{2}, \sqrt{3})= \Q(\sqrt{2}+\sqrt{3})$ Show that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal.   Proof. It follows from $\sqrt{2}+\sqrt{3} \in \Q(\sqrt{2}, \sqrt{3})$ that we have $\Q(\sqrt{2}+\sqrt{3})\subset \Q(\sqrt{2}, \sqrt{3})$. To show the reverse inclusion, […]
  • Calculate Determinants of MatricesCalculate Determinants of Matrices Calculate the determinants of the following $n\times n$ matrices. \[A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots […]
  • Finite Order Matrix and its TraceFinite 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) […]
  • The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than UnknownsThe Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns Determine all possibilities for the number of solutions of each of the system of linear equations described below. (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution. (b) A homogeneous system of $5$ equations in $4$ unknowns and the […]
  • Finite Group and a Unique Solution of an EquationFinite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that \[b^m=a.\]   We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]
  • Find Values of $a, b, c$ such that the Given Matrix is DiagonalizableFind Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix \[A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}\] diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the […]

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