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

• 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$3or 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 […]