linear=algebra-eye-catch2

LoadingAdd to solve later

Linear algebra problems and solutions


LoadingAdd to solve later

More from my site

  • Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient MatrixSolve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix Consider the following system of linear equations \begin{align*} 2x+3y+z&=-1\\ 3x+3y+z&=1\\ 2x+4y+z&=-2. \end{align*} (a) Find the coefficient matrix $A$ for this system. (b) Find the inverse matrix of the coefficient matrix found in (a) (c) Solve the system using […]
  • Determinant of Matrix whose Diagonal Entries are 6 and 2 ElsewhereDeterminant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere Find the determinant of the following matrix \[A=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & 2 \\ 2 & 2 & 2 & 6 & 2 \\ 2 & 2 & 2 & 2 & 6 \end{bmatrix}.\] (Harvard University, Linear Algebra Exam […]
  • Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$ For each positive integer $n$, prove that the polynomial \[(x-1)(x-2)\cdots (x-n)-1\] is irreducible over the ring of integers $\Z$.   Proof. Note that the given polynomial has degree $n$. Suppose that the polynomial is reducible over $\Z$ and it decomposes as […]
  • The Ring $\Z[\sqrt{2}]$ is a Euclidean DomainThe Ring $\Z[\sqrt{2}]$ is a Euclidean Domain Prove that the ring of integers \[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\] of the field $\Q(\sqrt{2})$ is a Euclidean Domain.   Proof. First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$. We use the […]
  • A Matrix Representation of a Linear Transformation and Related SubspacesA Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by \[ T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.\] (a) Find a matrix $A$ such that […]
  • Every Ring of Order $p^2$ is CommutativeEvery Ring of Order $p^2$ is Commutative Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$. Then prove that $R$ is a commutative ring.   Proof. Let us consider the subset \[Z:=\{z\in R \mid zr=rz \text{ for any } r\in R\}.\] (This is called the […]
  • Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces.Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces. Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$. (a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$. (b) If $B_1$ is a […]
  • Determinant of a General Circulant MatrixDeterminant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]

Leave a Reply

Your email address will not be published. Required fields are marked *