problems-in-mathematics-welcome-eye-catch

LoadingAdd to solve later


LoadingAdd to solve later

Sponsored Links

More from my site

  • The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent FunctionThe Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers. (a) Prove that the map $\exp:\R \to \R^{\times}$ defined by \[\exp(x)=e^x\] is an injective group homomorphism. (b) Prove that […]
  • Sequence Converges to the Largest Eigenvalue of a MatrixSequence Converges to the Largest Eigenvalue of a Matrix Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$. Furthermore, suppose that \[|\lambda_1| > |\lambda_2| \geq \cdots \geq […]
  • Determine a Condition on $a, b$ so that Vectors are Linearly DependentDetermine a Condition on $a, b$ so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}\] be vectors in $\R^3$. Determine a […]
  • Diagonalizable Matrix with Eigenvalue 1, -1Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]
  • Rotation Matrix in the Plane and its Eigenvalues and EigenvectorsRotation Matrix in the Plane and its Eigenvalues and Eigenvectors Consider the $2\times 2$ matrix \[A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$.   (a) Find the characteristic polynomial of the matrix $A$. (b) Find the […]
  • Inequality Regarding Ranks of MatricesInequality Regarding Ranks of Matrices Let $A$ be an $n \times n$ matrix over a field $K$. Prove that \[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\] where $\rk(B)$ denotes the rank of a matrix $B$. (University of California, Berkeley, Qualifying Exam) Hint. Regard the matrix as a linear transformation $A: […]
  • Compute Power of Matrix If Eigenvalues and Eigenvectors Are GivenCompute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
  • Is the Set of All Orthogonal Matrices a Vector Space?Is the Set of All Orthogonal Matrices a Vector Space? An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$. Let $V$ be the vector space of all real $2\times 2$ matrices. Consider the subset \[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.