# Diagonalization-eye-catch

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• Any Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as $\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,$ where $c_1, c_2, c_3$ are […]
• How to Obtain Information of a Vector if Information of Other Vectors are Given Let $A$ be a $3\times 3$ matrix and let $\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.$ Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$. Then find […]
• A Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\bar{f}:H\to K$ as follows. For each […]
• Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018.   Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is $2018=2\cdot 1009.$ Here $2$ and $1009$ are […]
• Idempotent Linear Transformation and Direct Sum of Image and Kernel Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$. We assume that $A$ is idempotent, that is, $A^2=A$. Then prove that $\R^n=\im(T) \oplus \ker(T).$   Proof. To prove the equality $\R^n=\im(T) […] • The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal Let$f:R\to R'$be a ring homomorphism. Let$I'$be an ideal of$R'$and let$I=f^{-1}(I)$be the preimage of$I$by$f$. Prove that$I$is an ideal of the ring$R$. Proof. To prove$I=f^{-1}(I')$is an ideal of$R$, we need to check the following two […] • Every Ring of Order$p^2$is Commutative Let$R$be a ring with unit$1$. Suppose that the order of$R$is$|R|=p^2$for some prime number$p$. Then prove that$R$is a commutative ring. Proof. Let us consider the subset $Z:=\{z\in R \mid zr=rz \text{ for any } r\in R\}.$ (This is called the […] • Order of Product of Two Elements in a Group Let$G$be a group. Let$a$and$b$be elements of$G$. If the order of$a, b$are$m, n$respectively, then is it true that the order of the product$ab$divides$mn\$? If so give a proof. If not, give a counterexample.   Proof. We claim that it is not true. As a […]