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• The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism. Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$. Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.   Proof. As in the […]
• Properties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
• Find All Matrices Satisfying a Given Relation Let $a$ and $b$ be two distinct positive real numbers. Define matrices \[A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.\] Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a […]
• Compute Determinant of a Matrix Using Linearly Independent Vectors Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have \[A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […]
• Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.\] (b) Find the inverse matrix of $A$.   Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
• A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by \[I:=\{ f(x) \in R \mid f(1)=0\}.\] Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]
• Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let \[ \mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}\] be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]
• Union of Subspaces is a Subspace if and only if One is Included in Another Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.     Proof. If $W_1 \cup W_2$ is a subspace, then $W_1 \subset W_2$ or $W_2 \subset […]