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  • Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite FieldPolynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements. For any nonzero element $a\in \F_p$, prove that the polynomial \[f(x)=x^p-x+a\] is irreducible and separable over $F_p$. (Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […]
  • Differentiation is a Linear TransformationDifferentiation 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 […]
  • Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$ Let $T:\R^3 \to \R^2$ be a linear transformation such that \[ T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},\] where $\mathbf{e}_1, […]
  • True or False Problems on Midterm Exam 1 at OSU Spring 2018True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ […]
  • Find the Rank of a Matrix with a ParameterFind the Rank of a Matrix with a Parameter Find the rank of the following real matrix. \[ \begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},\] where $a$ is a real number.   (Kyoto University, Linear Algebra Exam) Solution. The rank is the number of nonzero rows of a […]
  • Prove that the Center of Matrices is a SubspaceProve that the Center of Matrices is a Subspace Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define \[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\] The set $W$ is called the center of $V$. Prove that $W$ is a subspace […]
  • Dot Product, Lengths, and Distances of Complex VectorsDot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors \[ \mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} . \] Suppose $\mathbf{w}_4$ is […]
  • Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent? Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\] in $C[-2\pi, 2\pi]$. Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly […]

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