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Problems and solutions in Linear Algebra


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  • Questions About the Trace of a MatrixQuestions About the Trace of a Matrix Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
  • Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]
  • How to Diagonalize a Matrix. Step by Step Explanation.How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
  • If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian GroupIf a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group Let $G$ be a group with identity element $e$. Suppose that for any non identity elements $a, b, c$ of $G$ we have \[abc=cba. \tag{*}\] Then prove that $G$ is an abelian group.   Proof. To show that $G$ is an abelian group we need to show that \[ab=ba\] for any […]
  • Prove that the Center of Matrices is a SubspaceProve that the Center of Matrices is a Subspace Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define \[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\] The set $W$ is called the center of $V$. Prove that $W$ is a subspace […]
  • The Column Vectors of Every $3\times 5$ Matrix Are Linearly DependentThe Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent. (b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.   Proof. (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly […]
  • Fundamental Theorem of Finitely Generated Abelian Groups and its applicationFundamental Theorem of Finitely Generated Abelian Groups and its application In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem. Problem. Let $G$ be a finite abelian group of order $n$. If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic […]
  • The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$ Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by \[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\] That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$. Prove that the […]

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