# vector-space

by Yu · Published · Updated

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- Special Linear Group is a Normal Subgroup of General Linear Group Let $G=\GL(n, \R)$ be the general linear group of degree $n$, that is, the group of all $n\times n$ invertible matrices. Consider the subset of $G$ defined by \[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\] Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that […]
- Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively. Solution. Let $G$ be any nontrivial group, and let […]
- If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that the nullity of $T$ is zero. If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a […]
- Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$ Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix. (UCB-University of California, Berkeley, […]
- Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent Suppose $M$ is an $n \times n$ upper-triangular matrix. If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent. Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries? Proof. […]
- Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$ Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$. (a) Show that $T$ is a linear transformation. (b) Find a matrix $A$ such that […]
- Ring of Gaussian Integers and Determine its Unit Elements Denote by $i$ the square root of $-1$. Let \[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\] be the ring of Gaussian integers. We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to \[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\] Here $\bar{\alpha}$ is the complex conjugate of […]
- Every Group of Order 20449 is an Abelian Group Prove that every group of order $20449$ is an abelian group. Outline of the Proof Note that $20449=11^2 \cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]