# pi2018

• Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that $T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […] • The Center of the Symmetric group is Trivial if n>2 Show that the center Z(S_n) of the symmetric group with n \geq 3 is trivial. Steps/Hint Assume Z(S_n) has a non-identity element \sigma. Then there exist numbers i and j, i\neq j, such that \sigma(i)=j Since n\geq 3 there exists another […] • 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 […]
• Vector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ $Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
• An Example of a Matrix that Cannot Be a Commutator Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.   Proof. Assume that $[A, B]=-I$. Then $ABA^{-1}B^{-1}=-I$ implies $ABA^{-1}=-B. […] • Determine a Value of Linear Transformation From \R^3 to \R^2 Let T be a linear transformation from \R^3 to \R^2 such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […] • 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 […]
• Linear Dependent/Independent Vectors of Polynomials Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]