📘 linear algebra

1. **State the problem:** Find the QR-decomposition of the matrix $$A=\begin{bmatrix}1 & 2 & 1 \\ 1 & 1 & 1 \\ 0 & 3 & 1\end{bmatrix}$$ using the Gram-Schmidt process. 2. **Recall

1. The problem is to find the determinant of the matrix $$\begin{pmatrix} x-2 & 0 & 0 \\ 0 & y & -x \\ -y & 0 & z \end{pmatrix}$$

1. **Problem Statement:** We are given matrices partitioned into LHS and RHS blocks, with the condition that the sum of the product of any LHS column and any RHS column equals zero

1. **Problem Statement:** We are given matrices on the left-hand side (LHS) and right-hand side (RHS) and asked to partition matrix $A$ into $U$ and $L^T$ according to the example

1. **State the problem:** We need to check if the transformation $$T(x_1,x_2,x_3) = (x_1 - x_2, x_2 - x_3, x_1)$$ is a linear transformation. 2. **Recall the definition of linear t

1. Diketahui matriks A = $$\begin{bmatrix}5 & 2 & 3 & 2 \\ -1 & 0 & 1 & 3 \\ 4 & 1 & -4 & -1 \\ 1 & 2 & -1 & 0\end{bmatrix}$$

1. Problem statement: Find the rank of the matrix below and determine the type of solution of $A\vec{x}=\vec{0}$ for the first matrix given. 2. Matrix to analyze.

1. Problem statement: Find the rank of the first 4x4 matrix and determine the type of solutions of the homogeneous system $Ax=0$ for that matrix. 2. Formula and rules: The rank of

1. **Problem Statement:** Given the matrix equation $A\vec{x} = \vec{0}$, where $A$ is a $4 \times 4$ matrix, find the Reduced Row Echelon Form (RREF) of $A$, determine its rank, a

1. **Problem Statement:** We have sensor data from 15 wearable sensors recorded every second for a day, resulting in millions of records. We want to apply Principal Component Analy

1. **Problem:** Find the inverse of the matrix $$A = \begin{pmatrix}1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 2\end{pmatrix}$$ and solve the system: $$\begin{cases} x + 2y + z = 2 \\ 2x +

1. **Problem Statement:** Given a linear operator $T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $T(3,1) = (2,-4)$ and $T(1,2) = (0,2)$, find $T(a,b)$ and $T(7,9)$. 2. **Recall:** A

1. The problem is to rearrange and modify the provided midterm exam notes on linear transformations, matrix representations, eigenvalues, and related linear algebra concepts to mak

1. The problem asks: What is the second diagonal? 2. In a square or rectangular matrix, the "second diagonal" usually refers to the secondary diagonal, also called the anti-diagona

1. **Problem:** Determine if the function $\langle u, v \rangle = 3u_1v_1 + 5u_2v_2$ defines a valid inner product on $\mathbb{R}^2$. 2. **Recall the axioms of an inner product:**

1. **State the problem:** We are given a transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $T(c,b) = (c + b, -b, ab + 1)$ and asked to determine if $T$ is a linear trans

1. **Stating the problem:** We are given two 3x3 matrices: $$A = \begin{bmatrix}4 & 6 & 7 \\ 3 & 4 & 5 \\ 2 & -2 & 3\end{bmatrix}, \quad B = \begin{bmatrix}-3 & 5 & 3 \\ 2 & 3 & 2

1. **Stating the problem:** We want to solve the system of linear equations using Jacobi and Gauss-Seidel iterations with initial guess $[x^{(0)}, y^{(0)}, z^{(0)}] = [1, 2, 3]$ an

1. **Stating the problem:** Calculate the determinant of the matrix $$A = \begin{pmatrix} 6 & -3 & 2 \\ 2 & -1 & 2 \\ -10 & 5 & 2 \end{pmatrix}$$

1. **Problem Statement:** Given matrix

1. **Stating the problem:** We have a vector space $M$ of matrices of the form $\begin{pmatrix} a & b \\ c & 0 \end{pmatrix}$ with the constraint $a - 2b + c = 0$.