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• Boolean Rings Do Not Have Nonzero Nilpotent Elements Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a Boolean ring.) (a) Prove that $x^n=x$ for any positive integer $n$. (b) Prove that $R$ does not have a nonzero nilpotent […]
• An Example of a Real Matrix that Does Not Have Real Eigenvalues Let \[A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}\] be a $2\times 2$ matrix, where $a, b$ are real numbers. Suppose that $b\neq 0$. Prove that the matrix $A$ does not have real eigenvalues.   Proof. Let $\lambda$ be an arbitrary eigenvalue of […]
• Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$.   Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […]
• Positive definite Real Symmetric Matrix and its Eigenvalues A real symmetric $n \times n$ matrix $A$ is called positive definite if \[\mathbf{x}^{\trans}A\mathbf{x}>0\] for all nonzero vectors $\mathbf{x}$ in $\R^n$. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive. (b) Prove that if […]
• Differentiation is a Linear Transformation Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients. (a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\] for any $f(x)\in […]
• The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$ Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$. Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector […]
• 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 […]
• Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$. Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same.   Proof. Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent. This […]