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Purdue University Abstract Algebra Exam Problems and Solutions


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  • Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$. (a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ […]
  • The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$ Let $p$ be a prime number. Let $G$ be a non-abelian $p$-group. Show that the index of the center of $G$ is divisible by $p^2$. Proof. Suppose the order of the group $G$ is $p^a$, for some $a \in \Z$. Let $Z(G)$ be the center of $G$. Since $Z(G)$ is a subgroup of $G$, the order […]
  • Find All the Eigenvalues and Eigenvectors of the 6 by 6 MatrixFind All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 10001 & 3 & 5 & 7 &9 & 11 \\ 1 & 10003 & 5 & 7 & 9 & 11 \\ 1 & 3 & 10005 & 7 & 9 & 11 \\ 1 & 3 & 5 & 10007 & 9 & 11 \\ 1 &3 & 5 & 7 & 10009 & 11 \\ 1 &3 & 5 & 7 & 9 & […]
  • Subspace Spanned By Cosine and Sine FunctionsSubspace Spanned By Cosine and Sine Functions Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$. Define the map $f:\R^2 \to \calF[0, 2\pi]$ by \[\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […]
  • The Column Vectors of Every $3\times 5$ Matrix Are Linearly DependentThe Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent. (b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.   Proof. (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly […]
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  • The Sum of Cosine Squared in an Inner Product SpaceThe Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
  • Matrices Satisfying $HF-FH=-2F$Matrices Satisfying $HF-FH=-2F$ Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation \[HF-FH=-2F.\] (a) Find the trace of the matrix $F$. (b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ […]

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