# Princeton-university-eye-catch

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• Dimension 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 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 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 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 $ba \neq 1$ 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 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 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 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 \\ […]