📘 linear algebra

1. **State the problem:** Find the angle between vectors $\mathbf{u} = [-1, 3, 4]$ and $\mathbf{w} = [-2, -1, 3]$. 2. **Formula used:** The angle $\theta$ between two vectors $\mat

1. **State the problem:** We need to find the magnitude of the vector $3\mathbf{u} - \mathbf{v} + 2\mathbf{w}$ where $\mathbf{u} = [-1, 3, 4]$, $\mathbf{v} = [2, 1, -1]$, and $\mat

1. **State the problem:** We are given a vector $w = [-2, -1, 3]$ and need to find the quantity $\left\| \frac{4}{5} w \right\|$. 2. **Recall the formula:** The norm (or length) of

1. **Problem 15(a):** Solve the system $$\begin{cases} 2x - 3y = 1 \\ 6x - 9y = 3 \end{cases}$$

1. **Problem Statement:** Check whether the following sets form subspaces of their respective vector spaces.

1. **State the problem:** We have a population vector $\bar{P}_k = \begin{bmatrix} R_k \\ S_k \end{bmatrix}$ representing rats ($R_k$) and skunks ($S_k$) at year $k$. The populatio

1. **Problem Statement:** We are given a matrix $A$ of size $2025 \times 2025$ and asked if there exists another matrix $B$ of the same size such that $$AB - BA = I_{2025 \times 20

1. Misalkan matriks $B = \begin{bmatrix} 3 & -2 & 0 \\ -2 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$. Kita diminta menentukan determinan $\det(B)$ menggunakan operasi baris elementer. 2.

1. **State the problem:** Determine which of the given transformations are linear transformations. 2. **Recall the definition of a linear transformation:** A transformation $T$ is

1. **Problem Statement:** Given the matrix $$[S_{ij}] = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 3 & 0 & 3 \end{bmatrix}$$ and the vector $$[a_i] = \begin{bmatrix} 1 \\ 2 \\ 3 \en

1. **Problem Statement:** Given the matrix $$S = \begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & 2 \\ 3 & 0 & 3\end{bmatrix}$$ and the vector $$a = \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix},$$ e

1. **Problem Statement:** We are given a 3x3 matrix \(M_1 = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\) with determinant \(\det(M_1) = 8\). We want to find

1. **Problem Statement:** Find all vectors $\mathbf{v} = (x, y)$ that are orthogonal to $\mathbf{u} = (-7, 8)$. This means the dot product $\mathbf{u} \cdot \mathbf{v} = 0$. 2. **F

1. **Problem Statement:** We are given the augmented matrix of a linear system: $$\begin{bmatrix} 1 & 2 & -5 & -6 & 0 & | & 5 \\ 0 & 1 & -6 & -3 & 0 & | & 2 \\ 0 & 0 & 0 & 0 & 1 &

1. **Problem Statement:** Given a 3x3 matrix $B$ such that $B^2 = B$, determine which statements about $B$ must be true. 2. **Key Property:** The equation $B^2 = B$ means $B$ is id

1. **State the problem:** We have a set of vectors $W$ defined as $$\begin{bmatrix} a - 4b \\ 6 \\ 6a + b \\ -a - b \end{bmatrix}$$

1. The problem asks about the goal of the forward elimination steps in the Gaussian elimination method. 2. Gaussian elimination is a method used to solve systems of linear equation

1. The problem asks for the maximum possible number of leading 1's in the reduced row echelon form (RREF) of a matrix $A$ that is $5 \times 3$ (5 rows and 3 columns). 2. The leadin

1. **Problem Statement:** Given two square matrices $A$ and $B$ of order $n \times n$, find the expression for $(A - B)(A - B)$. 2. **Formula and Rules:** When multiplying matrices

1. **Stating the problem:** We have the system of linear equations: $$\begin{cases}

1. We are given matrices $AB = \begin{bmatrix}5 & 4 \\ -2 & 3\end{bmatrix}$ and $B = \begin{bmatrix}7 & 3 \\ 2 & 1\end{bmatrix}$, and we need to find matrix $A$. 2. The problem is