# Johns-Hopkins-exam-eye-catch

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• Linearly Independent/Dependent Vectors Question Let $V$ be an $n$-dimensional vector space over a field $K$. Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$. Are the following vectors linearly independent? $\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, […] • Are these vectors in the Nullspace of the Matrix? Let A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}. For each of the following vectors, determine whether the vector is in the nullspace \calN(A). (a) \begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix} […] • Every Cyclic Group is Abelian Prove that every cyclic group is abelian. Proof. Let G be a cyclic group with a generator g\in G. Namely, we have G=\langle g \rangle (every element in G is some power of g.) Let a and b be arbitrary elements in G. Then there exists […] • True or False: (A-B)(A+B)=A^2-B^2 for Matrices A and B Let A and B be 2\times 2 matrices. Prove or find a counterexample for the statement that (A-B)(A+B)=A^2-B^2. Hint. In general, matrix multiplication is not commutative: AB and BA might be different. Solution. Let us calculate (A-B)(A+B) as […] • Matrix Representations for Linear Transformations of the Vector Space of Polynomials Let P_2(\R) be the vector space over \R consisting of all polynomials with real coefficients of degree 2 or less. Let B=\{1,x,x^2\} be a basis of the vector space P_2(\R). For each linear transformation T:P_2(\R) \to P_2(\R) defined below, find the matrix representation […] • If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian? Let R be a commutative ring with 1. Suppose that the localization R_{\mathfrak{p}} is a Noetherian ring for every prime ideal \mathfrak{p} of R. Is it true that A is also a Noetherian ring? Proof. The answer is no. We give a counterexample. Let […] • A Group Homomorphism and an Abelian Group Let G be a group. Define a map f:G \to G by sending each element g \in G to its inverse g^{-1} \in G. Show that G is an abelian group if and only if the map f: G\to G is a group homomorphism. Proof. (\implies) If G is an abelian group, then f […] • The Set of Vectors Perpendicular to a Given Vector is a Subspace Fix the row vector \mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}, and let \R^3 be the vector space of 3 \times 1 column vectors. Define \[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.$ Prove that $W$ is a vector subspace of $\R^3$.   […]