📘 linear algebra

1. **State the problem:** Find the inverse of the matrix $$\mathbf{A}=\begin{bmatrix}2 & -17 & 11 \\ -11 & -7 & 0 \\ 3 & -2 & 1\end{bmatrix}.$$\n\n2. **Recall the formula for the i

1. **State the problem:** Find the inverse of the matrix $$A=\begin{bmatrix}2 & -17 & 1 \\ 1 & -1 & 1 \\ -7 & 0 & 3 \end{bmatrix}$$. 2. **Recall the formula:** The inverse of a mat

1. **State the problem:** Given matrices $$A = \begin{pmatrix}4 & 2 \\ 5 & 3\end{pmatrix}, \quad B = \begin{pmatrix}-5 & -2 \\ 4 & 1\end{pmatrix}$$

1. **Problem:** Solve the system using Gauss-Jordan Elimination. Given system:

1. **State the problem:** Solve the system of linear equations using Gauss elimination with partial pivoting: $$\begin{cases} x_1 - 3x_2 + x_3 = 9 \\ 2x_1 + x_2 - x_3 = 8 \\ 3x_1 -

1. **State the problem:** We are given matrix $$A = \begin{bmatrix}-6 & -9 & -8 \\ 2 & 9 & 6 \\ 0 & 1 & -1 \end{bmatrix}$$ and asked to find the third row of the adjugate matrix $$

1. **State the problem:** We need to find the $(1,2)$-entry of matrix $A$ given that $$(2A^{-1}+I)^{-1} = \begin{bmatrix}3 & 4 \\ 5 & 7\end{bmatrix}.$$ 2. **Rewrite the equation:**

1. **Problem:** Given matrices \( A = \begin{bmatrix}4 & -6 \\ -2 & 3\end{bmatrix} \), \( B = \begin{bmatrix}7 & -8 \\ -4 & 5\end{bmatrix} \), and a 2x2 matrix \( X \), if \( AX =

1. **Problem statement:** Given the vector $$x=\begin{bmatrix}\frac{15}{7} \\ \frac{16}{7}\end{bmatrix}$$, find the transformation matrix for expanding it on the x-axis by 4 units

1. **State the problem:** We have a vector $\mathbf{x} = \begin{bmatrix} \frac{11}{7} \\ \frac{18}{11} \end{bmatrix}$ in $\mathbb{R}^2$ and two linear transformations: a rotation $

1. **State the problem:** Prove that the sum $W_1 + W_2$ of two subspaces $W_1$ and $W_2$ is a direct sum if and only if their intersection is the zero vector space, i.e., $W_1 \ca

1. **Problem Statement:** Prove that if $W_1$ and $W_2$ are subspaces of a vector space $V$, then their intersection $W_1 \cap W_2$ is also a subspace of $V$. 2. **Recall the defin

1. **Problem statement:** Given matrix $$A = \begin{pmatrix} -1 & -2 & -5 + \alpha \\ 1 & -4 & 1 \\ 1 & -1 & 3 - \alpha \end{pmatrix}$$ and vector $$b = \begin{pmatrix} -3 \\ -3 \\

1. **Problem Statement:** Write each point A, B, C, D as a sum of scalar multiples of vectors $v_1$ and $v_2$.

1. **State the problem:** We want to find the values of $\omega$ such that the determinant of the matrix $$\begin{bmatrix} \omega - 1 & -1 & -2 \\ 0 & \omega - 2 & 2 \\ 0 & 0 & \om

1. **State the problem:** Find all values of $\omega$ such that the determinant of the matrix $$\begin{bmatrix} \omega - 1 & -1 & -2 \\ 0 & \omega - 2 & 2 \\ 0 & 0 & \omega - 3 \en

1. The problem is to understand what matrices are and how they are used. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. **Problem:** Use the Gram-Schmidt process to transform the basis \(v_1 = \begin{pmatrix}6 \\ 3 \\ 6\end{pmatrix}, v_2 = \begin{pmatrix}5 \\ 4 \\ 3\end{pmatrix}, v_3 = \begin{pma

1. **Stating the problem:** We are given a system of linear equations in matrix form: $$\begin{bmatrix} \theta_1 \\ v_1 \\ \theta_2 \\ v_2 \end{bmatrix} = \begin{bmatrix} 833.54 &

1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $\mathbf{\Phi}(t)$) for a given system. 2. **Formula and explana

1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $cb(t)$) for a given system. 2. **Formula and explanation:** The