# Pythagorean triple 2017

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• 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 […]
• Find an Orthonormal Basis of the Given Two Dimensional Vector Space Let $W$ be a subspace of $\R^4$ with a basis $\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.$ Find an orthonormal basis of $W$. (The Ohio State […]
• True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because […]
• Matrix Operations with Transpose Calculate the following expressions, using the following matrices: $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a) $A B^\trans + \mathbf{v} […] • A ring is Local if and only if the set of Non-Units is an Ideal A ring is called local if it has a unique maximal ideal. (a) Prove that a ring$R$with$1$is local if and only if the set of non-unit elements of$R$is an ideal of$R$. (b) Let$R$be a ring with$1$and suppose that$M$is a maximal ideal of$R$. Prove that if every […] • Linear Dependent/Independent Vectors of Polynomials Let$p_1(x), p_2(x), p_3(x), p_4(x)$be (real) polynomials of degree at most$3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At$1$each of the polynomials has the value$0$. Namely$p_i(1)=0$[…] • A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues Let$T:\R^2 \to \R^2$be a linear transformation and let$A$be the matrix representation of$T$with respect to the standard basis of$\R^2$. Prove that the following two statements are equivalent. (a) There are exactly two distinct lines$L_1, L_2$in$\R^2$passing through […] • Find all Column Vector$\mathbf{w}$such that$\mathbf{v}\mathbf{w}=0$for a Fixed Vector$\mathbf{v}$Let$\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all$3 \times 1$column vectors$\mathbf{w}$such that$\mathbf{v} \mathbf{w} = 0$. Solution. Let$\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}\$. Then we want \[\mathbf{v} […]