# linear-algebra-eyecatch

• Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions $e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$. Hint. Consider a linear combination $a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.$ […]
• All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix Let $A$ and $B$ be an $n \times n$ matrices. Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$. Then prove that each eigenvector of $A$ is an eigenvector of $B$. (It could be that each eigenvector is an eigenvector for […]
• The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.   Definition (Nonsingular Matrix). By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to […]
• Two Quotients Groups are Abelian then Intersection Quotient is Abelian Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups. Then show that the group $G/(K \cap N)$ is also an abelian group.   Hint. We use the following fact to prove the problem. Lemma: For a […]
• If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent Let $V$ be a subspace of $\R^n$. Suppose that $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}$ is a spanning set for $V$. Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.   We give two proofs. The essential ideas behind […]
• Any Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as $\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,$ where $c_1, c_2, c_3$ are […]
• Every Prime Ideal of a Finite Commutative Ring is Maximal Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$. Proof. We give two proofs. The first proof uses a result of a previous problem. The second proof is self-contained. Proof 1. Let $I$ be a prime ideal […]
• If the Quotient Ring is a Field, then the Ideal is Maximal Let $R$ be a ring with unit $1\neq 0$. Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$. (Do not assume that the ring $R$ is commutative.)   Proof. Let $I$ be an ideal of $R$ such that \[M \subset I \subset […]