Event_f_definition

• A Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]
• Linear Properties of Matrix Multiplication and the Null Space of a Matrix Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$.   Hint. Recall that the null space of an […]
• Subspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
• How to Obtain Information of a Vector if Information of Other Vectors are Given Let $A$ be a $3\times 3$ matrix and let $\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.$ Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$. Then find […]
• The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$ Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in […] • Equivalent Conditions to be a Unitary Matrix A complex matrix is called unitary if$\overline{A}^{\trans} A=I$. The inner product$(\mathbf{x}, \mathbf{y})$of complex vector$\mathbf{x}$,$\mathbf{y}$is defined by$(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector […] • Nilpotent Element a in a Ring and Unit Element$1-ab$Let$R$be a commutative ring with$1 \neq 0$. An element$a\in R$is called nilpotent if$a^n=0$for some positive integer$n$. Then prove that if$a$is a nilpotent element of$R$, then$1-ab$is a unit for all$b \in R$. We give two proofs. Proof 1. Since$a$[…] • Diagonalize the$2\times 2$Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix $A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.$ (a) Find the eigenvalues of$A$. (b) For each eigenvalue of$A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix$A\$ by a unitary matrix. Namely, find a diagonal matrix […]