📘 linear algebra

1. **State the problem:** Find the eigenvalues of the matrix $$A = \begin{pmatrix} 2 & 4 \\ 1 & 5 \end{pmatrix}$$. 2. **Formula and explanation:** Eigenvalues $$\lambda$$ satisfy t

1. **State the problem:** We are given six matrices A, B, C, D, E, and F and asked to find: (a) $A + B$

1. **Stating the problem:** We want to understand what vectors are and see an example with a drawing. 2. **Definition:** A vector is a quantity that has both magnitude (length) and

1. **State the problem:** We are given matrices with dimensions:

1. **Stating the problem:** We have matrices $A$, $B = I_3 + tA$, and $C = I_3 + aA$ where $I_3$ is the $3 \times 3$ identity matrix, and $a,t \in \mathbb{R}$. We want to find $a$

1. **Problem statement:** (i) Show that for any real number $x$ and vectors $u$ and $v$, the equality

1. **State the problem:** Calculate the determinant of the matrix $$A = \begin{pmatrix} 2 & 1 & -1 \\ 0 & 4 & 3 \\ -1 & 6 & 0 \end{pmatrix}$$ and then find its inverse $$A^{-1}$$.

1. **Stating the problem:** Determine which statements about matrices are always true and for which values of $a$ and $b$ the given system has more than one solution. 2. **Matrix s

1. Problem: Determine which statement about matrices is ALWAYS TRUE. 2. Analyze each option:

1. **State the problem:** We need to find the 2x2 matrix \( T = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) that transforms the point \((2,1)\) to \((1,4)\) and the point \((1,

1. **Problem statement:** We are given a linear code $C$ with a spanning set $\{11110111, 01100011, 00110010, 11001010, 01010001\}$. We need to find a basis for $C$. 2. **Recall:**

1. **State the problem:** We are given a transition matrix of the form $$\begin{bmatrix}-a-m & m & a \\ m & -m & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ and we want to analyze or solve for

1. **Problem statement:** Löse Aufgabe 12, das heißt, stelle einen der Vektoren a, b, c oder d als Linearkombination der anderen drei dar.

1. **Problem Statement:** You want to understand how to represent 2D affine transformations, which combine a 2x2 invertible matrix and a translation vector, as different groups and

1. **Stating the problem:** We are given the matrix

1. The problem asks to find a single elementary row operation that creates a 1 in the upper left corner of the given augmented matrix without creating fractions in the first row. 2

1. The problem is to find matrix $X$ such that $$X - \begin{bmatrix}1 & 7 \\ 3 & -2 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 7 \\ 3 & -2 \\ 0 & 1\end{bmatrix}$$

1. **State the problem:** We are given the system of linear equations: $$\begin{cases} x_1 + x_2 - 5x_3 = -8 \\ 5x_1 + 4x_2 - 5x_3 = -8 \end{cases}$$

1. **State the problem:** Solve the system of linear equations for $x_1$, $x_2$, and $x_3$: $$x_1 + x_2 - 5x_3 = -8$$

1. **State the problem:** Determine if the matrix $$\begin{pmatrix}-1 & 0.5 \\ 4 & -2 \\ 4 & 1\end{pmatrix}$$ has full rank. 2. **Recall the definition:** The rank of a matrix is t

1. Problem: Gegeben ist die quadratische Matrix $$A = \begin{pmatrix} 1 & -2 & 0 & 3 & -4 \\ -2 & 3 & 2 & 0 & 5 \\ 2 & -2 & -3 & 5 & -5 \\ 2 & -3 & -3 & -1 & 3 \\ 2 & 0 & -4 & -4 &