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• True or False Quiz About a System of Linear Equations (Purdue University Linear Algebra Exam)   Which of the following statements are true? (a) A linear system of four equations in three unknowns is always inconsistent. (b) A linear system with fewer equations than unknowns must have infinitely many solutions. (c) […]
• Complement of Independent Events are Independent Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are independent as well. Solution. Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that \[P(E) = P(E \cap F) + P(E […]
• Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ be an $2\times 2$ matrix. Express the eigenvalues of $A$ in terms of the trace and the determinant of $A$.   Solution. Recall the definitions of the trace and determinant of $A$: \[\tr(A)=a+d \text{ and } […]
• Submodule Consists of Elements Annihilated by Some Power of an Ideal Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$. Let $M'$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$. Prove that $M'$ is a submodule of […]
• True of False Problems on Determinants and Invertible Matrices Determine whether each of the following statements is True or False. (a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$. (b) If the characteristic polynomial of an $n \times n$ matrix $A$ […]
• Sequences Satisfying Linear Recurrence Relation Form a Subspace Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by \[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of […]
• Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$ Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.   Proof. Since $V$ is an $n$-dimensional vector space, it has a basis \[B=\{\mathbf{v}_1, \dots, […]
• 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 […]