# Harvard-University-exam-eye-catch

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• Pullback 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 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 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 $A, B$ 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 $A = […] • How to Calculate and Simplify a Matrix Polynomial Let$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 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? 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$A-aI$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 […]