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• If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$ Let $N$ be a normal subgroup of a group $G$. Suppose that $G/N$ is an infinite cyclic group. Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.   Hint. Use the fourth (or Lattice) isomorphism theorem. Proof. […]
• 7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
• Can We Reduce the Number of Vectors in a Spanning Set? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?   Solution. Yes, in general, $S_2$ can be a spanning set. As an […]
• Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ Determine whether there exists a nonsingular matrix $A$ if \[A^4=ABA^2+2A^3,\] where $B$ is the following matrix. \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\] If such a nonsingular matrix $A$ exists, find the inverse […]
• A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero.   Definition (Injective, One-to-One Linear Transformation). A linear […]
• 10 True or False Problems about Basic Matrix Operations Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don't lose a point if any of these is your exam problems. (These are actual exam […]
• Find Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
• An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […]