• Group Homomorphism, Preimage, and Product of Groups Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism. Put $N=\ker(f)$. Then show that we have $f^{-1}(f(H))=HN.$   Proof. $(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$. It follows that there exists $h\in H$ […]
• 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 […]
• The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers. Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive? If so, prove it. Otherwise, give a counterexample.   Solution. The […]
• Complement of Independent Events are Independent Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are independent as well. Solution. Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that $P(E) = P(E \cap F) + P(E […] • Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors (a) Let $A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}$ be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$. (Such a matrix is called (right) stochastic matrix (also termed […]
• 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 $a,b,c, d$ […]
• Projection to the subspace spanned by a vector Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$. (a) Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$. (b) Find a basis for the image subspace of $T$. (c) Find […]