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  • Determine linear transformation using matrix representationDetermine 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 GeneratedThe 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 VectorsDot 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 $A-aI$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 $A^3=O$.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 IsomorphicMultiplicative 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?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 SubspaceThe 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 $U$ […]

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