# UFD

• 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 […]
• Overall Fraction of Defective Smartphones of Three Factories A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]
• 12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) $S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}$ in […]
• True or False: Eigenvalues of a Real Matrix Are Real Numbers Answer the following questions regarding eigenvalues of a real matrix. (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers. (b) Find the eigenvalues of the matrix $B=\begin{bmatrix} -2 & […] • A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let A be an n\times n real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an n \times n real matrix B such that B^2=A. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […] • Explicit Field Isomorphism of Finite Fields (a) Let f_1(x) and f_2(x) be irreducible polynomials over a finite field \F_p, where p is a prime number. Suppose that f_1(x) and f_2(x) have the same degrees. Then show that fields \F_p[x]/(f_1(x)) and \F_p[x]/(f_2(x)) are isomorphic. (b) Show that the polynomials […] • Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ […]
• Finite Order Matrix and its Trace Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that (a) $|\tr(A)|\leq n$. (b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$. (c) $\tr(A)=n$ if and only if $A=I_n$. Proof. (a) […]