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- A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors Show that the set \[S=\{1, 1-x, 3+4x+x^2\}\] is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less. Proof. We know that the set $B=\{1, x, x^2\}$ is a basis for the vector space $P_2$. With respect to this basis $B$, the coordinate […]
- Nontrivial Action of a Simple Group on a Finite Set Let $G$ be a simple group and let $X$ be a finite set. Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$. Then show that $G$ is a finite group and the order of $G$ divides $|X|!$. Proof. Since $G$ acts on $X$, it […]
- True or False: Eigenvalues of a Real Matrix Are Real Numbers Answer the following questions regarding eigenvalues of a real matrix. (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers. (b) Find the eigenvalues of the matrix \[B=\begin{bmatrix} -2 & […]
- A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$ Let $R$ be a ring with $1$. Suppose that $a, b$ are elements in $R$ such that \[ab=1 \text{ and } ba\neq 1.\] (a) Prove that $1-ba$ is idempotent. (b) Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$. (c) Prove that the ring $R$ has infinitely many […]
- Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why. (b) Are the vectors \[ […]
- Eigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]
- A Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that \[f(n)=an\] for any integer $n$. Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
- If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that $A$ is a real $n\times n$ matrix. (a) Is it true that $A$ must commute with its transpose? (b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set. Is it true that the rows of $A$ must also form an orthonormal set? (University of […]