# top10mathproblems2017

by Yu ·

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### More from my site

- There is Exactly One Ring Homomorphism From the Ring of Integers to Any Ring Let $\Z$ be the ring of integers and let $R$ be a ring with unity. Determine all the ring homomorphisms from $\Z$ to $R$. Definition. Recall that if $A, B$ are rings with unity then a ring homomorphism $f: A \to B$ is a map […]
- Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$. Solution. We give several examples. The key facts are: An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain. An ideal $I$ of […]
- Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients. Let $T: P_3 \to V$ be the linear transformation defined by \[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix} a_0+a_2 & […]
- Every Maximal Ideal of a Commutative Ring is a Prime Ideal Let $R$ be a commutative ring with unity. Then show that every maximal ideal of $R$ is a prime ideal. We give two proofs. Proof 1. The first proof uses the following facts. Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […]
- Linear Properties of Matrix Multiplication and the Null Space of a Matrix Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$. Hint. Recall that the null space of an […]
- Finite Order Matrix and its Trace Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that (a) $|\tr(A)|\leq n$. (b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$. (c) $\tr(A)=n$ if and only if $A=I_n$. Proof. (a) […]
- Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal Let $R$ be a commutative ring. Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$. Proof. $(\implies)$: If $R$ is a field, then $\{0\}$ is a maximal ideal Suppose that $R$ is a field and let $I$ be a non zero ideal: \[ \{0\} […]
- A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. Let \[\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, […]