# linear-algebra-eyecatch

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- Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors) Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation \[a_{k+2}-5a_{k+1}+3a_{k}=0\] for $k=1, 2, \dots$. Let $T$ be […]
- Vector Form for the General Solution of a System of Linear Equations Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general […]
- If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group Let $G$ be a group with identity element $e$. Suppose that for any non identity elements $a, b, c$ of $G$ we have \[abc=cba. \tag{*}\] Then prove that $G$ is an abelian group. Proof. To show that $G$ is an abelian group we need to show that \[ab=ba\] for any […]
- A Group is Abelian if and only if Squaring is a Group Homomorphism Let $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$. Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism. Proof. $(\implies)$ If $G$ is an abelian group, then $f$ is a homomorphism. Suppose that […]
- Diagonalize the 3 by 3 Matrix Whose Entries are All One Diagonalize the matrix \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{bmatrix}.\] Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
- Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let \[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\] be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$. Proof. […]
- A Subgroup of the Smallest Prime Divisor Index of a Group is Normal Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$. Then prove that any subgroup of index $p$ is a normal subgroup of $G$. Hint. Consider the action of the group $G$ on the left cosets $G/H$ by left […]
- Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$. We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]