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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 […]
• Determinant/Trace and Eigenvalues of a Matrix Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]
• A Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]
• Given the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is given by \[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\] Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]
• Ideal Quotient (Colon Ideal) is an Ideal Let $R$ be a commutative ring. Let $S$ be a subset of $R$ and let $I$ be an ideal of $I$. We define the subset \[(I:S):=\{ a \in R \mid aS\subset I\}.\] Prove that $(I:S)$ is an ideal of $R$. This ideal is called the ideal quotient, or colon ideal.   Proof. Let $a, […]
• Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
• Sequences Satisfying Linear Recurrence Relation Form a Subspace Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by \[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of […]
• Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism. Prove that we have an isomorphism of groups: \[G \cong \ker(f)\times \Z.\]   Proof. Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]