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• A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by \[I:=\{ f(x) \in R \mid f(1)=0\}.\] Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]
• Are Coefficient Matrices of the Systems of Linear Equations Nonsingular? (a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular? (b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system […]
• Find the Conditional Probability About Math Exam Experiment A researcher conducted the following experiment. Students were grouped into two groups. The students in the first group had more than 6 hours of sleep and took a math exam. The students in the second group had less than 6 hours of sleep and took the same math exam. The pass […]
• True or False Quiz About a System of Linear Equations (Purdue University Linear Algebra Exam)   Which of the following statements are true? (a) A linear system of four equations in three unknowns is always inconsistent. (b) A linear system with fewer equations than unknowns must have infinitely many solutions. (c) […]
• A Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by \[ T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.\] (a) Find a matrix $A$ such that […]
• Matrix $XY-YX$ Never Be the Identity Matrix Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that \[XY-YX=I.\]   Hint. Suppose that such matrices exist and consider the trace of the matrix $XY-YX$. Recall that the trace of […]
• $\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]
• Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression \[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\] which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]