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Vector Space Problems and Solutions


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  • The Intersection of Two Subspaces is also a SubspaceThe Intersection of Two Subspaces is also a Subspace Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$. Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.   Definition (Intersection). Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]
  • The Center of a p-Group is Not TrivialThe Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial.   Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
  • Every Ideal of the Direct Product of Rings is the Direct Product of IdealsEvery Ideal of the Direct Product of Rings is the Direct Product of Ideals Let $R$ and $S$ be rings with $1\neq 0$. Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.   Proof. Let $K$ be an ideal of the direct product $R\times […]
  • $p$-Group Acting on a Finite Set and the Number of Fixed Points$p$-Group Acting on a Finite Set and the Number of Fixed Points Let $P$ be a $p$-group acting on a finite set $X$. Let \[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \] The prove that \[|X^P|\equiv |X| \pmod{p}.\]   Proof. Let $\calO(x)$ denote the orbit of $x\in X$ under the action of the group $P$. Let […]
  • Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure? If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module. Can this action be extended to make $M$ into a $\Q$-module?   Proof. In general, we cannot extend a $\Z$-module into a $\Q$-module. We give a counterexample. Let $M=\Zmod{2}$ be the order […]
  • A Square Root Matrix of a Symmetric Matrix with Non-Negative EigenvaluesA Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […]
  • Find a Matrix that Maps Given Vectors to Given VectorsFind a Matrix that Maps Given Vectors to Given Vectors Suppose that a real matrix $A$ maps each of the following vectors \[\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \] into the […]
  • Linear Algebra Midterm 1 at the Ohio State University (3/3)Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […]

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