nilpotent-matrix

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Nilpotent Matrix Problems and Solutions


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  • Cosine and Sine Functions are Linearly IndependentCosine and Sine Functions are Linearly Independent Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.   Proof. Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]
  • Is the Determinant of a Matrix Additive?Is the Determinant of a Matrix Additive? Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that \[\det(A+B)=\det(A)+\det(B)?\] If so, then give a proof. If not, then give a counterexample.   Solution. We claim that the statement is false. As a counterexample, […]
  • Submodule Consists of Elements Annihilated by Some Power of an IdealSubmodule Consists of Elements Annihilated by Some Power of an Ideal Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$. Let $M'$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$. Prove that $M'$ is a submodule of […]
  • Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials?Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials? 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 $T : \mathrm{P}_4 \rightarrow \mathrm{P}_{4}$ be the map defined by, for $f \in \mathrm{P}_4$, \[ […]
  • The Centralizer of a Matrix is a SubspaceThe Centralizer of a Matrix is a Subspace Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define \[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$. Prove that $W$ is a subspace of $V$.   Proof. First we check that the zero […]
  • Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-AxisLinear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis. Determine the formula for the function $F$ and prove that $F$ is a linear transformation.   Solution 1. Let $\begin{bmatrix} x \\ y […]
  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct EigenvaluesOrthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$. (Nagoya University, Linear Algebra Final Exam Problem)   Hint. Two […]

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