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Problems and solutions in Linear Algebra


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  • A Group Homomorphism is Injective if and only if MonicA Group Homomorphism is Injective if and only if Monic Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$. Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]
  • If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd OrderIf a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order Let $G$ be a finite group of order $2n$. Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup. Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities […]
  • Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a MatrixRow Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.\] (a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$. (b) Find a basis for the null space of $A$. (c) Find a basis for the range of $A$ that […]
  • Diagonalizable by an Orthogonal Matrix Implies a Symmetric MatrixDiagonalizable 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 […]
  • Find All Values of $x$ such that the Matrix is InvertibleFind All Values of $x$ such that the Matrix is Invertible Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if \[ A= \begin{bmatrix} 1 & 0 & c \\ 0 & a & -b \\ -1/a & x & x^{2} \end{bmatrix} . \]   Solution. We know that $A$ is invertible precisely when […]
  • How to Find Eigenvalues of a Specific Matrix.How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]
  • Find a Basis of the Eigenspace Corresponding to a Given EigenvalueFind a Basis of the Eigenspace Corresponding to a Given Eigenvalue Let \[A=\begin{bmatrix} 1 & 2 & 1 \\ -1 &4 &1 \\ 2 & -4 & 0 \end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$. Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$. (The Ohio State University, Linear Algebra Final Exam […]
  • Invertible Matrix Satisfying a Quadratic PolynomialInvertible Matrix Satisfying a Quadratic Polynomial Let $A$ be an $n \times n$ matrix satisfying \[A^2+c_1A+c_0I=O,\] where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? […]

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