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• Subspace Spanned By Cosine and Sine Functions Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$. Define the map $f:\R^2 \to \calF[0, 2\pi]$ by \[\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […]
• Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$. Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$. Proof. Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$. Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]
• The Inverse Matrix of an Upper Triangular Matrix with Variables Let $A$ be the following $3\times 3$ upper triangular matrix. \[A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},\] where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find […]
• Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent Prove that any set of vectors which contains the zero vector is linearly dependent.   Solution. Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set. Then we have the non-trivial linear combination \[1 \cdot […]
• Invertible Matrix Satisfying a Quadratic Polynomial Let $A$ be an $n \times n$ matrix satisfying \[A^2+c_1A+c_0I=O,\] where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? […]
• Problems and Solutions About Similar Matrices Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix. Prove the following statements. (a) If $A$ is similar to $B$, then $B$ is similar to $A$. (b) $A$ is similar to itself. (c) If $A$ is similar to $B$ and $B$ […]
• Describe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix \[A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.\]   Solution. By definition, the range $\calR(A)$ of the matrix $A$ is given […]
• Expectation, Variance, and Standard Deviation of Bernoulli Random Variables A random variable $X$ is said to be a Bernoulli random variable if its probability mass function is given by \begin{align*} P(X=0) &= 1-p\\ P(X=1) & = p \end{align*} for some real number $0 \leq p \leq 1$. (1) Find the expectation of the Bernoulli random variable $X$ […]