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• Find the Largest Prime Number Less than One Million. Find the largest prime number less than one million. What is a prime number? A natural number is called a "prime number" if it is only divisible by $1$ and itself. For example, $2, 3, 5, 7$ are prime numbers, although the numbers $4,6,9$ are not. The prime numbers have always […]
• A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$. Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module. Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.     Definition (Irreducible module). An […]
• Union of Two Subgroups is Not a Group Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$. (a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$. (b) Prove that a group cannot be written as the union of two proper […]
• Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that \[T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […]
• A Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
• Abelian Groups and Surjective Group Homomorphism Let $G, G'$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G'$. Show that if $G$ is an abelian group, then so is $G'$.   Definitions. Recall the relevant definitions. A group homomorphism $f:G\to G'$ is a map from $G$ to $G'$ […]
• Find Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
• Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations Here is a very short true or false problem. Select either True or False. Then click "Finish quiz" button. You will be able to see an explanation of the solution by clicking "View questions" button.