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• Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials Let $A$ be a $2\times 2$ real symmetric matrix. Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$.   Proof. Let $A=\begin{bmatrix} a& b \\ c& d \end{bmatrix}$. Then […]
• A Ring is Commutative if Whenever $ab=ca$, then $b=c$ Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$. Then prove that $R$ is a commutative ring.   Proof. Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$. Consider the […]
• Interchangeability of Limits and Probability of Increasing or Decreasing Sequence of Events A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition \[E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.\] Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition \[E_1 […]
• Determine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?   Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
• The Vector Space Consisting of All Traceless Diagonal Matrices Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero. That is, \begin{equation*} V:=\left\{ A=\begin{bmatrix} a_{11} & 0 & \dots & 0 \\ 0 &a_{22} & \dots & 0 \\ 0 & 0 & \ddots & \vdots \\ 0 & 0 & \dots & […]
• Elementary Questions about a Matrix Let \[A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.\] (a) What is the size of the matrix $A$? (b) What is the third column of $A$? (c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$. […]
• Idempotent Linear Transformation and Direct Sum of Image and Kernel Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$. We assume that $A$ is idempotent, that is, $A^2=A$. Then prove that \[\R^n=\im(T) \oplus \ker(T).\]   Proof. To prove the equality $\R^n=\im(T) […]
• 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 […]