More from my site

• How to Find the Determinant of the $3\times 3$ Matrix Find the determinant of the matix \[A=\begin{bmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 102 & 103 & 104 \end{bmatrix}.\]   Solution. Note that the determinant does not change if the $i$-th row is added by a scalar multiple of the $j$-th row if $i \neq […]
• 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 […]
• Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices (a) Prove that each complex $n\times n$ matrix $A$ can be written as \[A=B+iC,\] where $B$ and $C$ are Hermitian matrices. (b) Write the complex matrix \[A=\begin{bmatrix} i & 6\\ 2-i& 1+i \end{bmatrix}\] as a sum $A=B+iC$, where $B$ and $C$ are Hermitian […]
• If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.     Hint. Recall the basic properties of transpose […]
• Taking the Third Order Taylor Polynomial is a Linear Transformation The space $C^{\infty} (\mathbb{R})$ is the vector space of real functions which are infinitely differentiable. Let $T : C^{\infty} (\mathbb{R}) \rightarrow \mathrm{P}_3$ be the map which takes $f \in C^{\infty}(\mathbb{R})$ to its third order Taylor polynomial, specifically defined […]
• Short Exact Sequence and Finitely Generated Modules Let $R$ be a ring with $1$. Let \[0\to M\xrightarrow{f} M' \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}\] be an exact sequence of left $R$-modules. Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M'$ is also finitely generated.   […]
• Symmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$.   Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]
• Group Homomorphism, Preimage, and Product of Groups Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism. Put $N=\ker(f)$. Then show that we have \[f^{-1}(f(H))=HN.\]   Proof. $(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$. It follows that there exists $h\in H$ […]