📘 linear algebra

1. Let's start by defining what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. 2. A rectangular matrix is any matrix where the number of rows

1. Let's start by stating the problem: Understanding what a matrix is and how it works. 2. A matrix is a rectangular array of numbers arranged in rows and columns. For example, a m

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. A matrix is a rectangular array of numbers arranged in rows and col

1. **State the problem:** We need to evaluate the determinant of the matrix $$\begin{pmatrix}3 & -2 & 2 \\ 1 & 2 & -3 \\ 4 & 1 & 2\end{pmatrix}$$ using Laplace expansion along the

1. **State the problem:** We are given the system of linear equations: $$\begin{cases} x_1 + 3x_2 - 2x_3 = 6 \\ 4x_1 + 5x_2 + 2x_3 = 3 \\ x_1 + 3x_2 + 4x_3 = 3 \end{cases}$$

1. **State the problem:** We need to evaluate the determinant of the matrix $$\begin{pmatrix}3 & -2 & 2 \\ 1 & 2 & -3 \\ 4 & 1 & 2\end{pmatrix}$$ using Laplace expansion along the

1. **Problem Statement:** Find the fundamental matrix $\Phi(t)$ for the adjoint system of the linear system $\dot{x} = Ax$, where $$A = \begin{bmatrix} 1 & 3 & 8 \\ -2 & 2 & 1 \\ -

1. **Problem Statement:** Find a spanning set for the nullspace of the matrix $$A = \begin{bmatrix}-3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1 \\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}$

1. **Problem Statement:** We are given two 3x3 upper triangular matrices: $$A = \begin{bmatrix} a & 1 & 3 \\ 0 & e & 4 \\ 0 & 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} \psi &

1. **Problem Statement:** Given matrices $$A = \begin{bmatrix}-1 & 7 & -1 \\ 0 & 1 & 0 \\ 0 & 15 & -2\end{bmatrix}$$

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged in two rows and two columns

1. **Stating the problem:** We have four clinics each requiring a combination of four drugs (A, B, C, D) to meet their total treatment units. The goal is to find the allocation val

1. The problem is to express a system or equation in matrix form. 2. Matrix form typically means writing a system of linear equations as $AX = B$, where $A$ is the coefficient matr

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged as:

1. **State the problem:** Find the value of $k$ such that vectors $U = (2, 3k, -4, 1, 5)$ and $V = (6, -1, 3, 7, 2k)$ are orthogonal. 2. **Recall the definition of orthogonal vecto

1. **State the problem:** Solve the system of equations by calculating the inverse of the coefficient matrix using elementary row operations. The system is:

1. **Problem Statement:** Find the inverse of the matrices $$A = \begin{bmatrix}1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8\end{bmatrix}$$

1. **State the problem:** Prove that the vectors $\mathbf{u} = \langle 1, 1, 0 \rangle$, $\mathbf{v} = \langle 0, 1, 1 \rangle$, and $\mathbf{w} = \langle 1, 0, 1 \rangle$ are line

1. The problem involves understanding key concepts in linear algebra including vector spaces, subspaces, linear dependence and independence, basis, dimension, the four fundamental

1. **State the problem:** We are asked to find the product of two matrices: Matrix A (1x4): $$\begin{bmatrix}-3 & -4 & 1 & 3\end{bmatrix}$$

1. Асуудлыг тодорхойлж эхэлье: Хоёр вектор буюу хэрчмийг координатын хавтгай дээр байрлуулж, тэдгээрийн огтлолцлын бичлэгийг олох. 2. Хэрчмүүдийг координатын хавтгай дээр илэрхийлэ