# problems-in-mathematics-welcome-eye-catch

by Yu ·

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- A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero. Definition (Injective, One-to-One Linear Transformation). A linear […]
- Vector Space of Polynomials and a Basis of Its Subspace Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […]
- Ascending Chain of Submodules and Union of its Submodules Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain \[N_1 \subset N_2 \subset \cdots\] of submodules of $M$. Prove that the union \[\cup_{i=1}^{\infty} N_i\] is a submodule of $M$. Proof. To simplify the notation, let us […]
- Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
- A Prime Ideal in the Ring $\Z[\sqrt{10}]$ Consider the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\] and its ideal \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\] Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$. Definition of a prime ideal. An ideal $P$ of a ring $R$ is […]
- If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam) Proof. To show that $H$ is a normal subgroup of […]
- Matrix $XY-YX$ Never Be the Identity Matrix Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that \[XY-YX=I.\] Hint. Suppose that such matrices exist and consider the trace of the matrix $XY-YX$. Recall that the trace of […]
- The Determinant of a Skew-Symmetric Matrix is Zero Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd. Definition (Skew-Symmetric) A matrix $A$ is called skew-symmetric if $A^{\trans}=-A$. Here $A^{\trans}$ is the transpose of $A$. Proof. Properties of […]