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  • Dimension of the Sum of Two SubspacesDimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that \[\dim(U+V) \leq \dim(U)+\dim(V).\]   Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
  • Group of Invertible Matrices Over a Finite Field and its StabilizerGroup of Invertible Matrices Over a Finite Field and its Stabilizer Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number. Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]
  • Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field ExtensionIrreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension Prove that the polynomial \[f(x)=x^3+9x+6\] is irreducible over the field of rational numbers $\Q$. Let $\theta$ be a root of $f(x)$. Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.   Proof. Note that $f(x)$ is a monic polynomial and the prime […]
  • Group Homomorphism, Conjugate, Center, and Abelian groupGroup Homomorphism, Conjugate, Center, and Abelian group Let $G$ be a group. We fix an element $x$ of $G$ and define a map \[ \Psi_x: G\to G\] by mapping $g\in G$ to $xgx^{-1} \in G$. Then prove the followings. (a) The map $\Psi_x$ is a group homomorphism. (b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the […]
  • A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$ Let $R$ be a ring with $1$. Suppose that $a, b$ are elements in $R$ such that \[ab=1 \text{ and } ba\neq 1.\] (a) Prove that $1-ba$ is idempotent. (b) Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$. (c) Prove that the ring $R$ has infinitely many […]
  • Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly IndependentTwo Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively. Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are […]
  • Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative RingEvery Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$. Then prove that every prime ideal is a maximal ideal.   Hint. Let $R$ be a commutative ring with $1$ and $I$ be an ideal […]
  • Give a Formula for a Linear Transformation if the Values on Basis Vectors are KnownGive a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let \[ \mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}\] be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]

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