# top10mathproblems2017

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• A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space (a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix. Prove that $\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$. (b) Let $A$ be an $n\times n$ real matrix. Suppose […]
• Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces. Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$. (a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$. (b) If $B_1$ is a […]
• Show that the Given 2 by 2 Matrix is Singular Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$. (a) Show that $M$ is singular. (b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.   Solution. (a) Show […]
• A Positive Definite Matrix Has a Unique Positive Definite Square Root Prove that a positive definite matrix has a unique positive definite square root.   In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem. After the proof, several extra problems about square […]
• Is the Determinant of a Matrix Additive? Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that $\det(A+B)=\det(A)+\det(B)?$ If so, then give a proof. If not, then give a counterexample.   Solution. We claim that the statement is false. As a counterexample, […]
• Prove a Group is Abelian if $(ab)^2=a^2b^2$ Let $G$ be a group. Suppose that $(ab)^2=a^2b^2$ for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.   Proof. To prove that $G$ is an abelian group, we need $ab=ba$ for any elements $a, b$ in $G$. By the given […]
• If the Quotient by the Center is Cyclic, then the Group is Abelian Let $Z(G)$ be the center of a group $G$. Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian. Steps. Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$. Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$. Using […]
• Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$ Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying \[T\left(\, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\,\right)=\begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} […]