📘 linear algebra

1. **State the problem:** We need to solve the system of linear equations given by $$\begin{bmatrix}40 & -20 & -10 \\ -20 & 50 & -20 \\ -10 & -20 & 40\end{bmatrix} \begin{bmatrix}\

1. **State the problem:** We need to find the values of $\theta_1$, $\theta_2$, and $\theta_3$ that satisfy the matrix equation $$\begin{bmatrix} 40 & -20 & -10 \\ -20 & 50 & -20 \

1. **State the problem:** We are given the augmented matrix $$\begin{bmatrix} 1 & 0 & \frac{17}{10} & 44000 \\ 0 & 1 & -\frac{1}{10} & 3000 \\ 0 & 0 & 1 & -34000 \end{bmatrix}$$

1. **Problem Statement:** We have a bacteria culture with three types of bacteria (A, B, C) each requiring certain amounts of carbon, phosphate, and nitrogen daily. The total daily

1. **Problem Statement:** We are given a vector $v = (1,1)$ in $\mathbb{R}^2$. We want to show that reflecting $v$ about the $y$-axis produces the same result as rotating $v$ by 90

1. مسئله: یافتن ماتریس مجهول $X$ که با ضرب در ماتریس اول $A$، ماتریس دوم $B$ را به دست می‌دهد. 2. فرمول: اگر $A \times X = B$، برای یافتن $X$ باید از معکوس ماتریس $A$ استفاده کنیم،

1. مسئله: ماتریسی با سطر اول $[1, -1, -2]$ و سطر دوم $[3, 0, 1]$ داده شده است. 2. ماتریس را به صورت زیر می‌نویسیم:

1. مسئله: بررسی ماتریس ضرب شده با سطر اول $[2,1,3]$ و سطر دوم $[-1,1,2]$ است. 2. ماتریس ضرب شده باید به صورت صحیح نوشته شود تا عملیات ضرب ماتریسی قابل انجام باشد.

1. مسئله: ماتریسی به صورت $$\begin{bmatrix} 2 & 1 & 3 \\ -1 & 1 & 2 \end{bmatrix}$$ داریم که در ماتریس مجهولی ضرب می‌شود و حاصل آن ماتریس $$\begin{bmatrix} 1 & -1 & -2 \\ 3 & 0 & 1

1. **Problem 1:** Find the standard matrix $A$ representing the linear operator $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $L\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}

1. **State the problem:** We need to show that the determinant of the matrix $$\begin{vmatrix} 2 & 3 & -1 \\ 1 & 1 & 0 \\ 2 & -3 & 5 \end{vmatrix} = 0$$

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. This is a 2x2 matrix with elements \(a, b, c, d\).

1. **Problem Statement:** Find the eigenvalues and eigenvectors of the matrix $$A = \begin{bmatrix}5 & 7 \\ 3 & 4\end{bmatrix}$$ and interpret the geometric meaning of the eigenvec

1. **Problem Statement:** Given the matrix $$A = \begin{bmatrix}5 & 7 \\ 3 & 4\end{bmatrix}$$, we will perform matrix operations including addition, multiplication, and finding the

1. **Matrices Basics:** A matrix is a rectangular array of numbers arranged in rows and columns.

1. **Problem Statement:** We have a linear map $T : V \to W$ between $F$-vector spaces with $\dim V = \dim W = n$. Given a basis $\{\vec{b_1}, \ldots, \vec{b_n}\}$ of $V$, the stud

1. **Matrices Basics:** A matrix is a rectangular array of numbers arranged in rows and columns.

1. **State the problem:** We are given vectors $\mathbf{u} = [-1, 3, 4]$ and $\mathbf{v} = [2, 1, -1]$. We need to find the magnitude (or norm) of the vector $\mathbf{u} + \mathbf{

1. **State the problem:** We are given vectors $\mathbf{u} = [-1, 3, 4]$ and $\mathbf{v} = [2, 1, -1]$. We need to find the magnitude (or norm) of the vector $\mathbf{u} + \mathbf{

1. **State the problem:** Compute the linear combination $3\mathbf{u} + \mathbf{v} - \mathbf{w}$ where $\mathbf{u} = [1, 2, 1, 0]$, $\mathbf{v} = [-2, 0, 1, 6]$, and $\mathbf{w} =

1. The problem asks to compute the matrix $P_2$ and express its elements as fractions. 2. Typically, $P_2$ refers to the projection matrix onto the column space of a matrix $A$ wit