# Princeton-university-eye-catch

by Yu ·

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- A Subgroup of the Smallest Prime Divisor Index of a Group is Normal Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$. Then prove that any subgroup of index $p$ is a normal subgroup of $G$. Hint. Consider the action of the group $G$ on the left cosets $G/H$ by left […]
- Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent Suppose $M$ is an $n \times n$ upper-triangular matrix. If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent. Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries? Proof. […]
- Solving a System of Linear Equations By Using an Inverse Matrix Consider the system of linear equations \begin{align*} x_1&= 2, \\ -2x_1 + x_2 &= 3, \\ 5x_1-4x_2 +x_3 &= 2 \end{align*} (a) Find the coefficient matrix and its inverse matrix. (b) Using the inverse matrix, solve the system of linear equations. (The Ohio […]
- Idempotent Matrix and its Eigenvalues Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$. (a) Find a nonzero, nonidentity idempotent matrix. (b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$. (The Ohio State University, Linear Algebra Final Exam […]
- Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where \[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\] (a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for […]
- Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions? Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$. Solution. We claim that $T$ is not a […]
- Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
- Find Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. \[A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.\] Determine the values of $a$ so that the matrix $A$ is nonsingular. Solution. We use the fact that a matrix is nonsingular if and only if […]