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• Is the Map $T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3$ a Linear Transformation? Let $C ([0, 3] )$ be the vector space of real functions on the interval $[0, 3]$. Let $\mathrm{P}_3$ denote the set of real polynomials of degree $3$ or less. Define the map $T : C ([0, 3] ) \rightarrow \mathrm{P}_3 $ by \[T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) […]
• 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 […]
• The Group of Rational Numbers is Not Finitely Generated (a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated. (b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.   Proof. (a) Prove that the additive […]
• If There are 28 Elements of Order 5, How Many Subgroups of Order 5? Let $G$ be a group. Suppose that the number of elements in $G$ of order $5$ is $28$. Determine the number of distinct subgroups of $G$ of order $5$.   Solution. Let $g$ be an element in $G$ of order $5$. Then the subgroup $\langle g \rangle$ generated by $g$ is […]
• Orthonormal Basis of Null Space and Row Space Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$. (a) Find an orthonormal basis of the null space of $A$. (b) Find the rank of $A$. (c) Find an orthonormal basis of the row space of $A$. (The Ohio State University, Linear Algebra Exam […]
• The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate […]
• Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
• Given Eigenvectors and Eigenvalues, Compute a Matrix Product (Stanford University Exam) Suppose that $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is an eigenvector of a matrix $A$ corresponding to the eigenvalue $3$ and that $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ is an eigenvector of $A$ corresponding to the eigenvalue $-2$. Compute $A^2\begin{bmatrix} 4 […]