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• Solve a Linear Recurrence Relation Using Vector Space Technique Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be a subspace of $V$ defined by \[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\] Let $T$ be the linear transformation from […]
• If Two Matrices Have the Same Rank, Are They Row-Equivalent? If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If not, then provide a counterexample.   Solution. Having the same rank does not mean they are row-equivalent. For a simple counterexample, consider $A = […]
• Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors \[\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \\ 1 \end{bmatrix},\] respectively.   Solution. Suppose […]
• Lower and Upper Bounds of the Probability of the Intersection of Two Events Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that \[a \leq P(A \cap B) \leq b\] and $P(A \cap B)$ could […]
• Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$ Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.   Definition (Symmetric Matrix). A matrix $A$ is called symmetric if $A^{\trans}=A$. In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is […]
• Linearity of Expectations E(X+Y) = E(X) + E(Y) Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as \[E(X+Y) = E(X) + E(Y).\] Solution. The joint probability mass function of the discrete random variables $X$ and $Y$ is defined by \[p(x, y) = P(X=x, Y=y).\] Note that the […]
• Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• If Two Matrices are Similar, then their Determinants are the Same Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.   Proof. Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that \[S^{-1}AS=B\] by definition. Then we […]