# top10mathproblems2017

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- If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial Let $x, y$ be generators of a group $G$ with relation \begin{align*} xy^2=y^3x,\tag{1}\\ yx^2=x^3y.\tag{2} \end{align*} Prove that $G$ is the trivial group. Proof. Let $e$ be the identity element of $G$. We […]
- True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because […]
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
- Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times […]
- Prove that any Algebraic Closed Field is Infinite Prove that any algebraic closed field is infinite. Definition. A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$. Proof. Let $F$ be a finite field and consider the polynomial \[f(x)=1+\prod_{a\in […]
- Determinant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]
- Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have \[\phi(g^{-1})=\phi(g)^{-1}.\] Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
- Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices Let $A$ and $B$ be $n\times n$ matrices. Then prove that \[\calN(A)\cap \calN(B) \subset \calN(A+B),\] where $\calN(A)$ is the null space (kernel) of the matrix $A$. Definition. Recall that the null space (or kernel) of an $n \times n$ matrix […]