# nilpotent-matrix

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• Cosine 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? 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 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? 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 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 $x$-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 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 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 […]