# rss

by Yu · Published · Updated

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- Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by \[T\left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)= \begin{bmatrix} 3x_1+x_2 \\ x_1+3x_2 \end{bmatrix}.\] (a) Verify that the […]
- Ring Homomorphisms from the Ring of Rational Numbers are Determined by the Values at Integers Let $R$ be a ring with unity. Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$. Then prove that $f=g$. Proof. Let $a/b \in \Q$ be an arbitrary rational number with integers $a, b$. Then we […]
- If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that \[AB=O,\] where $O$ is the $3\times 3$ zero matrix. Proof. Since $A$ is singular, the equation $A\mathbf{x}=\mathbf{0}$ has a nonzero […]
- The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular. Definition (Nonsingular Matrix). By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to […]
- The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$ Suppose that $p$ is a prime number greater than $3$. Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$. (b) Determine the index $[G : S]$. (c) Assume […]
- Any Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series. Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]
- Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices Let $A$ and $B$ be $n\times n$ matrices. Then prove that \[\calN(A)\cap \calN(B) \subset \calN(A+B),\] where $\calN(A)$ is the null space (kernel) of the matrix $A$. Definition. Recall that the null space (or kernel) of an $n \times n$ matrix […]
- Group Homomorphism, Preimage, and Product of Groups Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism. Put $N=\ker(f)$. Then show that we have \[f^{-1}(f(H))=HN.\] Proof. $(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$. It follows that there exists $h\in H$ […]