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Johns Hopkins Linear Algebra Exam Problems and Solutions


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  • Linearly Independent/Dependent Vectors QuestionLinearly 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?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 AbelianEvery 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 $B$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 PolynomialsMatrix 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?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 GroupA 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 SubspaceThe 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 $\R^3$.   […]

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