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  • The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not IsomorphicThe Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.   Proof. We give three proofs. The first two proofs use only the properties of ring homomorphism. The third proof resort to the units of rings. If you are familiar with units of $\Z[x]$, then the […]
  • For What Values of $a$, Is the Matrix Nonsingular?For What Values of $a$, Is the Matrix Nonsingular? Determine the values of a real number $a$ such that the matrix \[A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}\] is nonsingular.   Solution. We apply elementary row operations and obtain: \begin{align*} A=\begin{bmatrix} 3 & 0 & a […]
  • Show the Subset of the Vector Space of Polynomials is a Subspace and Find its BasisShow the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]
  • Is an Eigenvector of a Matrix an Eigenvector of its Inverse?Is an Eigenvector of a Matrix an Eigenvector of its Inverse? Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not. (b) Is $3\mathbf{v}$ an […]
  • A Positive Definite Matrix Has a Unique Positive Definite Square RootA Positive Definite Matrix Has a Unique Positive Definite Square Root Prove that a positive definite matrix has a unique positive definite square root.   In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem. After the proof, several extra problems about square […]
  • If a Power of a Matrix is the Identity, then the Matrix is DiagonalizableIf a Power of a Matrix is the Identity, then the Matrix is Diagonalizable Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix. Show that the matrix $A$ is diagonalizable. Hint. Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]
  • The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chosen the numbers $7, 5, 3, 2, […]
  • Matrices Satisfying the Relation $HE-EH=2E$Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]

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