# Purdue-Algebra-Exam-eye-catch

• 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 $p^2$ 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 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 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 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 […] • Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective Let V be the vector space of all 2\times 2 real matrices and let P_3 be the vector space of all polynomials of degree 3 or less with real coefficients. Let T: P_3 \to V be the linear transformation defined by \[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix} a_0+a_2 & […] • The 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 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 $N$ […]