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  • How to Calculate and Simplify a Matrix PolynomialHow to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
  • Examples of Prime Ideals in Commutative Rings that are Not Maximal IdealsExamples of Prime Ideals in Commutative Rings that are Not Maximal Ideals Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.   Solution. We give several examples. The key facts are: An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain. An ideal $I$ of […]
  • The Sum of Cosine Squared in an Inner Product SpaceThe Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
  • Differentiating Linear Transformation is NilpotentDifferentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
  • The Powers of the Matrix with Cosine and Sine FunctionsThe Powers of the Matrix with Cosine and Sine Functions Prove the following identity for any positive integer $n$. \[\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos […]
  • Compute Determinant of a Matrix Using Linearly Independent VectorsCompute Determinant of a Matrix Using Linearly Independent Vectors Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have \[A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […]
  • Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional SpaceQuiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by \[W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\] (a) Which of the following vectors are in the subset […]
  • Find the Limit of a MatrixFind the Limit of a Matrix Let \[A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}\] be $3 \times 3$ matrix. Find \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear […]

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