# cody-slides

by Yu ·

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### More from my site

- Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions \[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$. Solution. The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given […]
- 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 […]
- Hyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors \[\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n\] satisfying the linear equation of the form \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]
- Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
- Vector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]
- If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix. Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]
- Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given […]
- A Group Homomorphism is Injective if and only if the Kernel is Trivial Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$. Definitions/Hint. We recall several […]