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• 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 $p_i(1)=0$ […]