📘 linear algebra

1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $\mathbf{\Phi}(t)$) for a given system. 2. **Formula and explana

1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $cb(t)$) for a given system. 2. **Formula and explanation:** The

1. **Problem statement:** We have a square system $A_{sq} x = b_{sq}$ where $$A_{sq} = \begin{bmatrix} 2 & 20 & 2 \\ 4 & 45 & 1.6 \\ 6 & 60 & 2 \end{bmatrix}, \quad b_{sq} = \begin

1. **Problem statement:** Find the transformation matrix $M$ for (a) a reflection in the line $y=\frac{\sqrt{3}}{3}x$ followed by a clockwise rotation of $30^\circ$ about the origi

1. **Problem:** Determine the solution type for the system with reduced row echelon form (RREF): $$\begin{bmatrix}1 & 0 & 0 & | & -2 \\ 0 & 1 & 0 & | & 3 \\ 0 & 0 & 0 & | & 0\end{b

1. **State the problem:** We are given a linear system:

1. **State the problem:** We are given matrices $$A = \begin{bmatrix}-5 & -1 & 5 \\ 3 & -3 & -8\end{bmatrix}$$

1. **State the problem:** We are given two matrices $A$ and $B$ and asked to perform matrix operations: $-3A$, $A - 3B$, and $4A + 5B$. 2. **Recall matrix operations:**

1. **Problem statement:** Solve the system of linear equations: $$\begin{cases}-4x_2 + x_3 + 2x_4 = 8 \\ 2x_1 + 3x_2 + x_3 - x_4 = 3 \\ x_1 - 2x_2 + 4x_3 + 3x_4 = 19 \end{cases}$$

1. **State the problem:** We are given a system of equations: $$

1. **State the problem:** We are given a system of equations: $$

1. The problem asks which variables are basic in the basic solution (D) from the given system and table. 2. Recall that in a system of linear equations with slack variables, a basi

1. **State the problem:** We have the system of equations: $$2x_1 + 3x_2 + s_1 = 27$$

1. The problem asks: In basic solution (B), which variables are basic? 2. The system of equations is:

1. The problem is to find the expression for $J^{-n}$ where $J$ is the matrix $$J = \begin{pmatrix}4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4\end{pmatrix}$$

1. **Problem statement:** Given the matrix $$J = \begin{pmatrix}4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4\end{pmatrix}$$, find the matrix $$J^{-n}$$ for a positive integer $$n$$. 2. **Ke

1. **Stating the problem:** We want to demonstrate the theorem that if the entries of any row (or column) of a determinant are multiplied by a nonzero number $k$ and added to the c

1. **State the problem:** We have the system of linear equations: $$\frac{1}{4}x - \frac{5}{4}y - \frac{5}{4}z = -10$$

1. **Problem Statement:** Given a technology matrix $A$ and a production schedule vector $X$, find the external demand vector $D$ such that $$X = AX + D.$$

1. Problem statement: Determine if the vector (1,7,23) can be a vector perpendicular to both $\vec{a} = (2,3,-1)$ and $\vec{b} = (5,-1,-2)$ in problem 7b. 2. Formula: To find a vec

1. **Problemstellung:** Warum wählt man in der Lösung für den Vektor \( \vec{c} = \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} \) den Wert \( c_1 = 3 \) aus?