# Field-theory-eye-catch

by Yu · Published · Updated

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- Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$. \[V:=\left\{ \quad\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \in \R^4 \quad \middle| \quad x_1-x_2+x_3-x_4=0 \quad\right\}.\] Find a basis of the subspace $V$ […]
- Eigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]
- Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
- Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer. Prove that the polynomial \[f(x)=x^n-t\] in the ring $S[x]$ is irreducible in $S[x]$. Proof. Consider the principal ideal $(t)$ generated by $t$ […]
- Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$? If so, prove it. Otherwise give a counterexample. Proof. We give a counterexample. Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]
- A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable. Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
- Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$. Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same. Proof. Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent. This […]
- 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 […]