📘 linear algebra

1. **State the problem:** We have the system of linear equations: $$x - ky + z = 0,$$

1. The problem is to represent an anticlockwise rotation of a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). 2. A 2D anticlockwise rotation by an angle \(\theta\) is pe

1. The problem asks us to find the matrix $C$ which results from applying matrix $A$ (anticlockwise rotation by 90°) followed by matrix $B$ (reflection in the line $y = -x$), i.e.,

1. The problem asks to determine the values of $k$ for which the system $$

1. The determinant of a 3x3 matrix $A=\begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}$ using Sarrus' rule is calculated as: $$\det(A) = aei + bfg + cdh - ceg - bdi -

1. Problem (a): Find the SVD of matrix $$A = \begin{pmatrix} -2 & 2 \\ 1 & 1 \end{pmatrix}$$. 2. Step 1: Compute $$A^TA$$ and $$AA^T$$.

1. **Problem Statement:** Find the inverse of the matrix $$A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0\end{bmatrix}.$$ Verify which given matrix is indeed $$A^{-1}$$. 2. *

1. **Vector Operations:** Given vectors \(\mathbf{u} = \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix}4 \\ 5 \\ 6\end{pmatrix}\). 2. **Vector sum \(\ma

1. مسئله اول: فرض کنید $A \in \mathbb{C}^{n \times n}$ و $AA^* = A^*A$. نشان دهید ماتریسی $U$ وجود دارد که $$U^*AU = D, \quad U^*U = UU^* = I,$$ که در آن $D$ ماتریس قطری شامل مقادی

1. We are given a matrix $$A=\begin{bmatrix}1&2&0&-1\\3&0&1&2\\2&-1&3&1\\1&4&2&0\end{bmatrix} \in \mathbb{R}^{4 \times 4}$$ and some properties about it and other matrices. 2. Firs

1. First, let's state the problem: We want to understand why the scalar $y^T X w$ equals its transpose $(X w)^T y$. 2. Note that $y^T$ is a row vector, $X$ is a matrix, and $w$ is

1. Let's start by stating the problem: We want to verify why $ (y - Xw)^T (y - Xw) = \|y\|^2 - 2y^T X w + w^T X^T X w $. 2. Recall that the squared norm of a vector $a$ is $a^T a$.

1. Let's state the problem: We want to understand why $$y^T A = (A^T y)^T$$ holds, where $y$ is a vector and $A$ is a matrix. 2. Recall that for a vector $y$ (column vector), $y^T$

1. Stating the problem: We want to diagonalize matrix $$A=\begin{bmatrix}1 & -6 & 4 \\ 0 & 4 & 2 \\ 0 & -6 & 3\end{bmatrix}$$. 2. Find the eigenvalues by solving $$\det(A-\lambda I

1. The problem asks to understand what a singular matrix is. 2. A matrix $A$ is called singular if it does not have an inverse.

1. The problem is to understand what a matrix is and its basic properties. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. The problem involves matrices multiplication followed by evaluating a scalar expression. 2. The first matrix is a 3x3: $$\begin{pmatrix} -2 & 0 & 5 \\ -1 & 3 & 6 \\ 0 & 1 & -1 \

1. **State the problem:** We have matrices $$A=\begin{pmatrix}2 & x \\ 3 & 1 \end{pmatrix}, B=\begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix}, C=\begin{pmatrix}3x+2 & 7 \\ 7-x & 7 \end

1. **State the problem:** We need to find the dimension of the space of all upper triangular matrices of order $n$. These are square matrices $A = (a_{ij})$ where $a_{ij} = 0$ for

1. You mentioned "the matrix u" but did not provide the matrix or the problem to solve. 2. Please provide the matrix or specify the problem so I can assist you step-by-step.

1. Stating the problem: Find the determinant of a 3\times3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \). 2. The determinant \( \det(A) \) of a