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Tokyo University Linear Algebra Exam Problems and Solutions


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  • How Many Solutions for $x+x=1$ in a Ring?How Many Solutions for $x+x=1$ in a Ring? Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation \[x+x=1 \] has more than one solutions $x\in R$?   Solution. We claim that there is at most one solution $x$ in the ring $R$. Suppose that we have two solutions $r, s \in R$. That is, we […]
  • Isomorphism Criterion of Semidirect Product of GroupsIsomorphism Criterion of Semidirect Product of Groups Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism. The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation \[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\] where $a_i […]
  • If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field. Let $R$ be a commutative ring with $1$. Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.   Proof. As the zero ideal $(0)$ of $R$ is a proper ideal, it is a prime ideal by assumption. Hence $R=R/\{0\}$ is an integral […]
  • The Product of Two Nonsingular Matrices is NonsingularThe Product of Two Nonsingular Matrices is Nonsingular Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem)   Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
  • Prove that the Center of Matrices is a SubspaceProve that the Center of Matrices is a Subspace Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define \[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\] The set $W$ is called the center of $V$. Prove that $W$ is a subspace […]
  • Determinant/Trace and Eigenvalues of a MatrixDeterminant/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 […]
  • Determine the Dimension of a Mysterious Vector Space From Coordinate VectorsDetermine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let $V$ be a vector space and $B$ be a basis for $V$. Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$. Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]
  • A Group of Linear FunctionsA Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]

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