# number2018

by Yu ·

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- Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix \[A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}\] diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem) Solution. Note that the […]
- Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$ Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace \[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\] Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
- Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]
- Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where \[\mathbf{a}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix} 1 \\ 1 […]
- A ring is Local if and only if the set of Non-Units is an Ideal A ring is called local if it has a unique maximal ideal. (a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$. (b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$. Prove that if every […]
- 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 […]
- All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$ Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$. Solution. Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$. Note that the linear […]
- Finite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that \[b^m=a.\] We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]