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  • Basis with Respect to Which the Matrix for Linear Transformation is DiagonalBasis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by \[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]
  • Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$ Let $A$ be the coefficient matrix of the system of linear equations \begin{align*} -x_1-2x_2&=1\\ 2x_1+3x_2&=-1. \end{align*} (a) Solve the system by finding the inverse matrix $A^{-1}$. (b) Let $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be the solution […]
  • Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a MatrixDetermine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. \[A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […]
  • The Rank of the Sum of Two MatricesThe Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that \[\rk(A+B) \leq \rk(A)+\rk(B).\] Proof. Let \[A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],\] where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
  • Galois Group of the Polynomial  $x^p-2$.Galois Group of the Polynomial $x^p-2$. Let $p \in \Z$ be a prime number. Then describe the elements of the Galois group of the polynomial $x^p-2$.   Solution. The roots of the polynomial $x^p-2$ are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1\] where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]
  • Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct ProductExample of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.   Solution. Let $G$ be any nontrivial group, and let […]
  • In which $\R^k$, are the Nullspace and Range Subspaces?In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.   Solution. For an $m \times n$ matrix $A$, the […]
  • Diagonalize a 2 by 2 Symmetric MatrixDiagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]

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