# Yu-Tsumura-small

by Yu · Published · Updated

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- The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers. Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive? If so, prove it. Otherwise, give a counterexample. Solution. The […]
- Fundamental Theorem of Finitely Generated Abelian Groups and its application In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem. Problem. Let $G$ be a finite abelian group of order $n$. If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic […]
- Find the Limit of a Matrix Let \[A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}\] be $3 \times 3$ matrix. Find \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear […]
- Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix \[A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.\] (a) Prove that the linear transformation […]
- The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$ Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in […]
- Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$. Solution. We […]
- If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix. Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$. Proof. Because $A$ has rank $n$, we know that the $n \times n$ […]
- $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$. Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]