📘 linear algebra

1. **Problem:** Find the eigenvalues and eigenvectors of a given square matrix $A$. 2. **Formula and Explanation:**

1. **State the problem:** We are given the matrix $$\begin{bmatrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix}$$

1. The problem is to find the trace of the matrix $CF - I$, where $C$ and $F$ are matrices and $I$ is the identity matrix. 2. Recall the definition of the trace: the trace of a mat

1. **State the problem:** Find the norm (or magnitude) of the vector $\mathbf{v} = (-1, 2, 4)$.\n\n2. **Formula:** The norm of a vector $\mathbf{v} = (v_1, v_2, v_3)$ in 3D space i

1. The problem asks to find the norm (or magnitude) of vector $v$ for part (a) where $v = (1, -1)$. 2. The norm of a vector $v = (x, y)$ in 2D is given by the formula:

1. **State the problem:** We want to simplify a large determinant into the product of two factors. 2. **Formula and rules:** The determinant of a matrix can sometimes be factored i

1. **State the problem:** We are given the matrix \(A = \begin{pmatrix} 3 & 1 & 3 & 4 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 5 & 3 \\ 0 & 0 & -3 & -1 \end{pmatrix}\) and need to find its Jo

1. **Problem statement:** Given the set $U \subset \mathbb{R}^4$ defined by the vectors $$

1. **State the problem:** Solve the system of linear equations using the inverse of the coefficient matrix. The system is:

1. **Problem:** Find the value of $a$ for which the matrix $$\begin{bmatrix} 2 & 0 & 1 \\ 5 & a & 3 \\ 0 & 3 & 1 \end{bmatrix}$$

1. The problem asks to rewrite the expression for $x^{-1}$ given as a matrix: $$x^{-1} = \begin{pmatrix}0 & \frac{1}{3} \\ \frac{1}{2} & -\frac{1}{6}\end{pmatrix}$$

1. **Problem statement:** Find a unitary matrix $U$ and an upper triangular matrix $P$ such that $U^* A U = P$ for the matrix $$A = \begin{bmatrix} -1 & 5 & 1 \\ -1 & 2 & 1 \\ 1 &

1. **Problem Statement:** Check whether the given matrix $$\begin{bmatrix}3 & 0 & 1 \\ 2 & 0 & 4 \\ 1 & 5 & 2\end{bmatrix}$$

1. **Problem statement:** We are given a 4x4 matrix, two matrices A (3x2) and B (2x3), and a system of four linear equations with variables $x_1, x_2, x_3, x_4$.

1. **Problem statement:** We have a matrix $P = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ representing a geometric transformation $U$.

1. The problem is to find the trace of a matrix.\n\n2. The trace of a square matrix is the sum of the elements on its main diagonal.\n\n3. If the matrix is \( A = \begin{bmatrix} a

1. **Problem Statement:** Solve the system of simultaneous linear equations using the matrix inverse method: $$\begin{cases} 2x + 3y - z = 1 \\ -x + 2y + z = 8 \\ x - 3y - 2z = -13

1. We are asked to find the determinant of the matrix: $$\begin{bmatrix} 5 & 8 & 14 \\ 6 & 12 & 13 \\ 8 & 5 & 6 \end{bmatrix}$$

1. **State the problem:** We want to solve the system $Ax=\bar{b}$ where $$A=\begin{bmatrix}4 & 2 & 0 \\ -8 & -2 & -7 \\ 13 & 11 & 21 \\ \end{bmatrix}, \quad \bar{b}=\begin{bmatrix

1. **Problem Statement:** Given matrices

1. **Problem Statement:** Understand the concepts of orthogonal and orthonormal vectors, orthogonal and orthonormal bases, orthogonal matrices, and their properties in linear algeb