nilpotent-matrix

• Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID (a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal. (b) Prove that a quotient ring of a PID by a prime ideal is a PID.   Proof. (a) Prove that every PID is a maximal ideal. Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]
• The Sum of Subspaces is a Subspace of a Vector Space Let $V$ be a vector space over a field $K$. If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset $W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}$ is a subspace of the vector space $V$.   Proof. We prove the […]
• Vector Space of Polynomials and a Basis of Its Subspace 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)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […] • Find All Values of$x$so that a Matrix is Singular Let $A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a$4\times 4$matrix. Find all values of$x$so that the matrix$A$is singular. Hint. Use the fact that a matrix is singular if and only […] • Find the Nullity of the Matrix$A+I$if Eigenvalues are$1, 2, 3, 4, 5$Let$A$be an$n\times n$matrix. Its only eigenvalues are$1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix$A+I_n$, where$I_n$is the$n\times n$identity matrix? (The Ohio State University, Linear Algebra Final Exam […] • The Rank of the Sum of Two Matrices Let$A$and$B$be$m\times n$matrices. Prove that $\rk(A+B) \leq \rk(A)+\rk(B).$ Proof. Let $A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],$ where$\mathbf{a}_i$and$\mathbf{b}_i$are column vectors of$A$and$B$, […] • If Two Subsets$A, B$of a Finite Group$G$are Large Enough, then$G=AB$Let$G$be a finite group and let$A, B$be subsets of$G$satisfying $|A|+|B| > |G|.$ Here$|X|$denotes the cardinality (the number of elements) of the set$X$. Then prove that$G=AB$, where $AB=\{ab \mid a\in A, b\in B\}.$ Proof. Since$A, B$[…] • Idempotent Matrix and its Eigenvalues Let$A$be an$n \times n$matrix. We say that$A$is idempotent if$A^2=A$. (a) Find a nonzero, nonidentity idempotent matrix. (b) Show that eigenvalues of an idempotent matrix$A$is either$0$or$1\$. (The Ohio State University, Linear Algebra Final Exam […]