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• Are These Linear Transformations? Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0 \end{bmatrix} ,\; S\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x+y […]
• Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions. $A=aP+(a+1)Q$ $P^2=P$ $Q^2=Q$ $PQ=O$ $QP=O$, where $O$ is the $2\times 2$ zero matrix. Then do the following problems. (a) Prove that […]
• 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 […]
• Determine a Condition on $a, b$ so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}\] be vectors in $\R^3$. Determine a […]
• The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$ Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by \[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\] That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$. Prove that the […]
• Any Automorphism of the Field of Real Numbers Must be the Identity Map Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.   Proof. We prove the problem by proving the following sequence of claims. Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]
• Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix Find all eigenvalues of the matrix \[A=\begin{bmatrix} 0 & i & i & i \\ i &0 & i & i \\ i & i & 0 & i \\ i & i & i & 0 \end{bmatrix},\] where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric […]
• 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 […]