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Linear Transformation problems and solutions


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  • 7 Problems on Skew-Symmetric Matrices7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
  • Group Homomorphism, Preimage, and Product of GroupsGroup 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 $h\in H$ […]
  • Rings $2\Z$ and $3\Z$ are Not IsomorphicRings $2\Z$ and $3\Z$ are Not Isomorphic Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.   Definition of a ring homomorphism. Let $R$ and $S$ be rings. A homomorphism is a map $f:R\to S$ satisfying $f(a+b)=f(a)+f(b)$ for all $a, b \in R$, and $f(ab)=f(a)f(b)$ for all $a, b \in R$. A […]
  • If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero MatrixIf a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix.   Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]
  • Invertible Matrix Satisfying a Quadratic PolynomialInvertible Matrix Satisfying a Quadratic Polynomial Let $A$ be an $n \times n$ matrix satisfying \[A^2+c_1A+c_0I=O,\] where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? […]
  • Vector Form for the General Solution of a System of Linear EquationsVector Form for the General Solution of a System of Linear Equations Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general […]
  • $p$-Group Acting on a Finite Set and the Number of Fixed Points$p$-Group Acting on a Finite Set and the Number of Fixed Points Let $P$ be a $p$-group acting on a finite set $X$. Let \[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \] The prove that \[|X^P|\equiv |X| \pmod{p}.\]   Proof. Let $\calO(x)$ denote the orbit of $x\in X$ under the action of the group $P$. Let […]
  • Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions?Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions? Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$.   Solution. We claim that $T$ is not a […]

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