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• Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
• Rank and Nullity of a Matrix, Nullity of Transpose Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$. The dimension of the nullspace of $A$ is called the nullity of $A$. Prove the followings. (a) $\calN(A)=\calN(A^{\trans}A)$. (b) $\rk(A)=\rk(A^{\trans}A)$.   Hint. For part (b), […]
• If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that \[AB=O,\] where $O$ is the $3\times 3$ zero matrix.   Proof. Since $A$ is singular, the equation $A\mathbf{x}=\mathbf{0}$ has a nonzero […]
• Sequences Satisfying Linear Recurrence Relation Form a Subspace Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by \[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of […]
• All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$ Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.   Solution. Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$. Note that the linear […]
• Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.   Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
• 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. From the relation (1), we […]
• 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} […]