# Yu-Tsumura-small

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• The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by $T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = […] • Find All Matrices B that Commutes With a Given Matrix A: AB=BA Let \[A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.$ Then (a) Find all matrices $B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}$ such that $AB=BA$. (b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that $AB=BA \text{ and } […] • The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let G be an abelian group. Let a and b be elements in G of order m and n, respectively. Prove that there exists an element c in G such that the order of c is the least common multiple of m and n. Also determine whether the statement is true if G is a […] • Every Group of Order 72 is Not a Simple Group Prove that every finite group of order 72 is not a simple group. Definition. A group G is said to be simple if the only normal subgroups of G are the trivial group \{e\} or G itself. Hint. Let G be a group of order 72. Use the Sylow's theorem and determine […] • Linear Dependent/Independent Vectors of Polynomials Let p_1(x), p_2(x), p_3(x), p_4(x) be (real) polynomials of degree at most 3. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At 1 each of the polynomials has the value 0. Namely p_i(1)=0 […] • Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors Let V be a subset of \R^4 consisting of vectors that are perpendicular to vectors \mathbf{a}, \mathbf{b} and \mathbf{c}, where \[\mathbf{a}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix} 1 \\ 1 […] • 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 […] • If a Group G Satisfies abc=cba then G is an Abelian Group Let G be a group with identity element e. Suppose that for any non identity elements a, b, c of G we have \[abc=cba. \tag{*}$ Then prove that $G$ is an abelian group.   Proof. To show that $G$ is an abelian group we need to show that $ab=ba$ for any […]