# linear-algebra-eyecatch

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• A Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]
• The Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that $\rk(A+B) \leq \rk(A)+\rk(B).$ Proof. Let $A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],$ where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
• Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. Let A be the matrix $\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.$ Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse. (The Ohio State University Linear Algebra […]
• The Order of $ab$ and $ba$ in a Group are the Same Let $G$ be a finite group. Let $a, b$ be elements of $G$. Prove that the order of $ab$ is equal to the order of $ba$. (Of course do not assume that $G$ is an abelian group.)   Proof. Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is, $(ab)^n=e, […] • Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring R is an ideal of R? If so, prove it. Otherwise give a counterexample. Proof. We give a counterexample. Let R be the noncommutative ring of 2\times 2 matrices with real […] • 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 […] • Are these vectors in the Nullspace of the Matrix? Let A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}. For each of the following vectors, determine whether the vector is in the nullspace \calN(A). (a) \begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix} […] • Find the Eigenvalues and Eigenvectors of the Matrix A^4-3A^3+3A^2-2A+8E. Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.$ Find the eigenvalues and the eigenvectors of the matrix $B=A^4-3A^3+3A^2-2A+8E.$ (Nagoya University Linear Algebra Exam Problem)   Hint. Apply the Cayley-Hamilton theorem. That is if $p_A(t)$ is the […]