# linear=algebra-eye-catch2

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• Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements Let $\F_3=\Zmod{3}$ be the finite field of order $3$. Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$. (a) Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have? (b) […]
• Linear Independent Continuous Functions Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set $S=\{ \sqrt{x}, x^2 \}$ in $C[3,10]$. Show that the set $S$ is linearly independent in $C[3,10]$.   Proof. Note that the zero vector […]
• If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$ Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.   Proof. As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$. Let us think backwards. Suppose that […]
• Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix $A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.$ (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
• Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like. Consider the matrix $A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).$ (a) Find the eigenvalues and corresponding eigenvectors of $A$. (b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose […]
• Rings $2\Z$ 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 […]
• Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let $W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}$ be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.   Proof. […]
• A Group Homomorphism is Injective if and only if the Kernel is Trivial Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.     Definitions/Hint. We recall several […]