# problems-in-mathematics-welcome-eye-catch

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• Determine linear transformation using matrix representation Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations. \begin{align*} T\left(\, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 0 \\ 1 […]
• The Group of Rational Numbers is Not Finitely Generated (a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated. (b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.   Proof. (a) Prove that the additive […]
• Dot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors $\mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} .$ Suppose $\mathbf{w}_4$ is […]
• Maximize the Dimension of the Null Space of $A-aI$ Let $A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.$ Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
• Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […] • Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic Let$\R^{\times}=\R\setminus \{0\}$be the multiplicative group of real numbers. Let$\C^{\times}=\C\setminus \{0\}$be the multiplicative group of complex numbers. Then show that$\R^{\times}$and$\C^{\times}$are not isomorphic as groups. Recall. Let$G$and$K$[…] • Are Linear Transformations of Derivatives and Integrations Linearly Independent? Let$W=C^{\infty}(\R)$be the vector space of all$C^{\infty}$real-valued functions (smooth function, differentiable for all degrees of differentiation). Let$V$be the vector space of all linear transformations from$W$to$W$. The addition and the scalar multiplication of$V$[…] • The Intersection of Two Subspaces is also a Subspace Let$U$and$V$be subspaces of the$n$-dimensional vector space$\R^n$. Prove that the intersection$U\cap V$is also a subspace of$\R^n$. Definition (Intersection). Recall that the intersection$U\cap V$is the set of elements that are both elements of$U\$ […]