📘 linear algebra

1. **Problem statement:** We are given a parametric vector equation \( \vec{x} = \begin{pmatrix}420 \\ -630 \\ 120 \end{pmatrix} + r \cdot \begin{pmatrix}40 \\ 50 \\ 11 \end{pmatri

1. **State the problem:** Determine if the vectors \(\vec{v}_1 = \begin{bmatrix} -16 \\ 8 \\ -4 \end{bmatrix}\), \(\vec{v}_2 = \begin{bmatrix} 16 \\ 20 \\ -4 \end{bmatrix}\), and \

1. **State the problem:** We want to verify if the matrix \(\begin{bmatrix} 2/17 & -5/7 \\ -1/17 & 6/7 \end{bmatrix}\) is the inverse of \(\begin{bmatrix} 6 & -5 \\ -1 & -2 \end{bm

1. **State the problem:** Determine if the subset $H$ of $V=\mathbb{R}^2$, consisting of all points in the first and third quadrants between the lines $y=2x$ and $y=\frac{x}{2}$, i

1. **Stating the problem:** We are asked to multiply the pseudo-inverse of matrix $A$ by matrix $B$, and then multiply matrix $A$ by its pseudo-inverse to verify if the result is t

1. **Problem statement:** Find the inverse of the 2x2 matrices \(A = \begin{bmatrix} 3 & -1 \\ -1 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} -3 & 2 \\ 6 & -4 \end{bmatrix}\) usi

1. **Stating the problem:** We have a transformation represented by the matrix $$\begin{pmatrix}1 & -1 \\ 0 & 2\end{pmatrix}$$.

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x_1 + x_2 = -5 \\ x_2 + x_3 = -2 \\ x_3 + x_4 = -1 \\ x_1 + x_4 = -4 \end{cases}$$

1. The problem asks: What is the pivot element? 2. In linear algebra, a pivot element is the first non-zero element in a row of a matrix after performing Gaussian elimination or ro

1. **State the problem:** We need to find matrix $A$ such that $$\left(3A + \begin{bmatrix}-1 & -2 \\ -5 & 4 \\ 2 & 4\end{bmatrix}\right)^T = \begin{bmatrix}-4 & 2 & 5 \\ -3 & 3 &

1. The problem involves understanding the matrix $A = \begin{bmatrix}1 & 7 & 6 \\ 5 & 6 & 1\end{bmatrix}$ and its reduced row echelon form (rref) $\begin{bmatrix}1 & 0 & -1 \\ 0 &

1. **State the problem:** We are given matrix $A = \begin{bmatrix}1 & 7 & 6 \\ 5 & 6 & 1\end{bmatrix}$ which corresponds to a system of linear equations. We need to write the syste

1. **Problem statement:** Given matrices \(A, B, C, D\), perform matrix operations and answer related questions. ---

1. **State the problem:** We are given the vector equation $$5\begin{pmatrix}7\\-\frac{1}{4}x\\12\end{pmatrix} = \frac{9}{2}\begin{pmatrix}y\\y\\y\end{pmatrix}$$ and need to find t

1. **Stating the problem:** We have a system of linear equations:

1. The problem is to determine which pairs of matrices are row equivalent. 2. Two matrices are row equivalent if one can be transformed into the other by a sequence of elementary r

1. The problem asks to perform the row operation $-7R_3 + 13R_2$ on the given matrix and find the resulting entries $a_i, b_i, c_i, d_i$ for rows $R_1, R_2, R_3$. 2. The original m

1. **State the problem:** We have two matrix transformations: matrix $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ transforms point $A$ to point $B$, and matrix $\begin{pmatrix}3 &

1. **Problem statement:** Check if the mapping \(F : \mathbb{R}^4 \to M(2 \times 2, \mathbb{R})\) defined by

1. The problem asks: A 2 \times 2 matrix with every entry equal to 1 is a ? (square matrix or multiplicative identity matrix). 2. First, let's understand the definitions:

1. **State the problem:** Find the inverse of matrix $$A = \begin{bmatrix}0 & -1 & 1 \\ 1 & 2 & 0 \\ -1 & 0 & -1\end{bmatrix}$$. 2. **Method:** Use row operations on the augmented