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• Find a Basis for the Range of a Linear Transformation of Vector Spaces of Matrices Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by $T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b - d & -3c \\ 2b - c […] • Any Finite Group Has a Composition Series Let G be a finite group. Then show that G has a composition series. Proof. We prove the statement by induction on the order |G|=n of the finite group. When n=1, this is trivial. Suppose that any finite group of order less than n has a composition […] • Diagonalize a 2 by 2 Matrix A and Calculate the Power A^{100} Let \[A=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}.$ (a) Find eigenvalues of the matrix $A$. (b) Find eigenvectors for each eigenvalue of $A$. (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]
• Linear Algebra Midterm 1 at the Ohio State University (2/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 2 and contains […]
• Equivalent Definitions of Characteristic Subgroups. Center is Characteristic. Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$. (a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$. (b) Prove that the center […]
• Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism. Prove that we have an isomorphism of groups: $G \cong \ker(f)\times \Z.$   Proof. Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]
• Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
• Compute $A^5\mathbf{u}$ Using Linear Combination Let \[A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } […]