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  • Pullback Group of Two Group Homomorphisms into a GroupPullback Group of Two Group Homomorphisms into a Group Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms. Define the subset $M$ of $G_1 \times G_2$ to be \[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\] Prove that $M$ is a subgroup of $G_1 \times G_2$.   […]
  • 7 Problems on Skew-Symmetric Matrices7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
  • If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero MatrixIf a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that \[AB=O,\] where $O$ is the $3\times 3$ zero matrix.   Proof. Since $A$ is singular, the equation $A\mathbf{x}=\mathbf{0}$ has a nonzero […]
  • If Two Matrices Have the Same Rank, Are They Row-Equivalent?If Two Matrices Have the Same Rank, Are They Row-Equivalent? If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If not, then provide a counterexample.   Solution. Having the same rank does not mean they are row-equivalent. For a simple counterexample, consider $A = […]
  • How to Calculate and Simplify a Matrix PolynomialHow to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
  • Projection to the subspace spanned by a vectorProjection 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 […]
  • Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix? Let $A$ be an $n \times n$ matrix. Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.   Solution. The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]
  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]

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