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  • The Centralizer of a Matrix is a SubspaceThe Centralizer of a Matrix is a Subspace Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define \[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$. Prove that $W$ is a subspace of $V$.   Proof. First we check that the zero […]
  • Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of SystemsDetermine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations. (a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$. (b) […]
  • No/Infinitely Many Square Roots of 2 by 2 MatricesNo/Infinitely Many Square Roots of 2 by 2 Matrices (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$. (b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root […]
  • Is the Quotient Ring of an Integral Domain still an Integral Domain?Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain?   Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
  • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
  • 12 Examples of Subsets that Are Not Subspaces of Vector Spaces12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) \[S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\] in […]
  • Is the Derivative Linear Transformation Diagonalizable?Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by \[ T( ax^2 + bx + c ) = 2ax + b . \] Is $T$ diagonalizable? If so, find a diagonal matrix which […]
  • Describe the Range of the Matrix Using the Definition of the RangeDescribe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix \[A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.\]   Solution. By definition, the range $\calR(A)$ of the matrix $A$ is given […]

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