# problems-in-mathematics-welcome-eye-catch

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• The 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 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 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, -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 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 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 $A: […] • Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let$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? 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 […]

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