📘 linear algebra

1. **Problem statement:** We want to generally prove and explain the orthogonality of vectors using the scalar product (dot product) and vector product (cross product). 2. **Defini

1. **Problem statement:** Given vectors $\vec{a} = \begin{pmatrix}2 \\ 4\end{pmatrix}$ and $\vec{b} = \begin{pmatrix}-1 \\ 3\end{pmatrix}$, draw the vectors $\vec{a}$, $\vec{b}$, $

1. **State the problem:** We are given four matrices: \(A\), \(B\), \(C\), and \(D\). We need to find the sum of the two square matrices among them. 2. **Identify the square matric

1. Problemet handlar om att bestämma arean av parallellogrammet som spänns upp av vektorerna $\mathbf{u} = (-4, 2, 3)$ och $\mathbf{v} = (4, -1, -3)$.\n\n2. Arean av parallellogram

1. **State the problem:** We are given a vector expression in terms of variables $x_0$, $x_5$, and $x_7$:

1. **State the problem:** Find the determinant of the matrix $$\begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix}$$

1. **Problem Statement:** We have the set $V = \mathbb{R}^3$ with vectors $v = (a_1, a_2, a_3)$ and $w = (b_1, b_2, b_3)$ where $a_i, b_i \in \mathbb{Z}$. The operations are define

1. **State the problem:** Find the characteristic polynomial of the matrix $$A = \begin{bmatrix} 4 & 0 & 10 \\ -3 & 1 & -10 \\ 0 & 1 & -2 \end{bmatrix}$$ using the variable $$\lamb

1. **Problem statement:** Find the characteristic polynomial of matrix \( A = \begin{bmatrix}-5 & 10 & -10 \\ -14 & 19 & -17 \\ -14 & 14 & -12 \end{bmatrix} \) using the variable \

1. **Problem Statement:** Encode the message "QUANTUM JUMP" using the encoding matrix \(A=\begin{pmatrix}1 & 2 & 1 \\ 2 & 3 & 1 \\ -2 & 0 & 1\end{pmatrix}\) with the letter-to-numb

1. **Problem Statement:** Determine the geometry of the systems of equations given by (a) and (b) using normal analysis. 2. **Recall:** The geometry of three planes in 3D can be:

1. **Problem Statement:** Determine if the statement "If $A$ is an $n \times n$ matrix with $n$ linearly independent columns, then $A^2$ is invertible" is true or false.

1. **Problem Statement:** We want to check if the equality $$(A + B)(A - B) = A^2 - B^2$$ holds for general $n \times n$ matrices $A$ and $B$. 2. **Recall the formula for differenc

1. **Problem Statement:** Given matrices $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}, \quad B = \begin{bmatrix}1 & 2 \\ -1 & 1\end{bmatrix}$$

1. **Stating the problem:** We are given the matrix $$A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$$ and need to find $$A^3$$, which means multiplying matrix $$A$$ by itself th

1. **Problem Statement:** Find the standard matrix $A$ of the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ such that $$

1. The problem is to describe vectors and illustrate some examples. 2. A vector is a quantity that has both magnitude (length) and direction.

1. **State the problem:** Given vectors $\mathbf{u} = (0, 4)$ and $\mathbf{v} = (3, 3)$, find the vector $4\mathbf{u} + 2\mathbf{v}$. 2. **Recall the formula:** Scalar multiplicati

1. The problem states: If the products $AB$ and $BA$ are defined, then $AB$ and $BA$ are square matrices. 2. To understand this, recall the rule for matrix multiplication: For two

1. **State the problem:** Find the inverse of the matrix $$\begin{bmatrix}4 & -3 \\ 12 & 3\end{bmatrix}$$ given that its determinant is 38. 2. **Recall the formula for the inverse

1. **State the problem:** We are given two matrices: $$A = \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}$$