📘 linear algebra

1. **State the problem:** We need to find the determinant of the adjoint of the matrix $$A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 2 \\ 0 & 4 & 5 \end{pmatrix}$$. 2. **Recall the fo

1. **Problem Statement:** Determine which of the given projection relationships among vectors $A$, $B$, $C$, $D$, and $E$ holds true. 2. **Recall the projection formula:** The proj

1. **Stating the problem:** We are asked to multiply two matrices $$A$$ and $$B$$ where

1. **Problem Statement:** Find the determinant of each given matrix. 2. **Formula and Rules:**

1. **State the problem:** We want to find the condition on $$\vec{b}$$ such that the system $$B\vec{v} = \vec{b}$$ is solvable, where $$B = \begin{bmatrix} 0 & 2 & 1 \\ 1 & 1 & 1 \

1. The problem is to understand the concept of matrices. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. **Problem Statement:** We are given a matrix $$A = \begin{bmatrix}4 & 1 \\ 2 & 3\end{bmatrix}$$ and asked to represent it graphically as a 2\times2 matrix labeled $A$ positioned

1. **Stating the problem:** We are given a matrix $$A = \begin{bmatrix} 4 & -2 & 7 \\ 5 & 3 & 1 \end{bmatrix}$$. 2. **Understanding the matrix:** This is a $$2 \times 3$$ matrix wi

1. Let's start by stating the problem: We want to understand what a matrix is and see an example. 2. A matrix is a rectangular array of numbers arranged in rows and columns. For ex

1. The problem is to provide an example of a matrix. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. **State the problem:** We want to find the transition matrix from basis $B$ to basis $\mathbf{B'}$ in $\mathbb{R}^3$. This matrix converts coordinates of vectors from the $B$-ba

1. **State the problem:** We want to find the transition matrix from basis $B$ to basis $\mathbf{B}'$ in $\mathbb{R}^3$.

1. The problem is to understand and explain Laplace's theorem. 2. Laplace's theorem, also known as the Laplace expansion, is a method to calculate the determinant of a square matri

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

1. \textbf{โจทย์:} ให้ A, B, C และ I เป็นเมตริกซ์ขนาด $4 \times 4$ โดยที่ $I$ เป็นเมตริกซ์เอกลักษณ์ \text{ให้} \det(-A)^4 = \det(4I), \quad \det(2C)^{-1} = 32, \quad \text{และ} \qu

1. **Nêu bài toán:** Cho ma trận $$A = \begin{pmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{pmatrix}$$ với các trị riêng $$\lambda_1 = \lambda_2 = -1$$ và $$\lambda_3 = 5$$. Tìm

1. **Đặt vấn đề:** Chúng ta cần xác định ma trận nào trong số các ma trận A, B, C, D không phải là ma trận chéo.

1. **Nêu bài toán:** Cho ma trận $$A = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$$, ta cần xác định ma trận này có chéo hóa được hay không và tìm các vectơ riêng. 2. **Công thức

1. **Nêu bài toán:** Cho ma trận $A$ và các trị riêng $\lambda_1=2$, $\lambda_2=4$, $\lambda_3=8$ cùng các vector riêng tương ứng $u_1=(1,1,0)$, $u_2=(1,-1,0)$, $u_3=(0,0,1)$. Yêu

1. **โจทย์:** กำหนดเมทริกซ์ $$A=\begin{bmatrix}5 & -3 \\ 2 & -1\end{bmatrix}$$ และ $$B=\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$$ a. หาค่า $$\mathrm{tr}(B^T A)$$

1. **Problem Statement:** Understand the null space of a matrix $A$ in linear algebra. 2. **Definition:** The null space (or kernel) of a matrix $A$ is the set of all vectors $\mat