# Linear-algebra-quiz-eye-catch

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- In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$. Solution. For an $m \times n$ matrix $A$, the […]
- If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$. Hint. It follows from […]
- Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or Less Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less. (a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$. (b) Write the polynomial $f(x) = 2 + 3x - x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 […]
- Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […]
- Characteristic of an Integral Domain is 0 or a Prime Number Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$. Definition of the characteristic of a ring. The characteristic of a commutative ring $R$ with $1$ is defined as […]
- The Matrix Exponential of a Diagonal Matrix For a square matrix $M$, its matrix exponential is defined by \[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\] Suppose that $M$ is a diagonal matrix \[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & […]
- Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if \[A_1^2+A_2^2+\cdots+A_m^2=\calO,\] where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$. Hint. Recall that a complex matrix $A$ is Hermitian if […]
- Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]