📘 linear algebra

1. **Problem statement:** Given matrix $$A=\begin{bmatrix}1 & 0 & 3 \\ 2 & 5 & 3 \\ 2 & 0 & 5\end{bmatrix}$$, find the element at the intersection of row 3 and column 2 of the inve

1. **Problem statement:** Find the inverse of matrix $$A = \begin{bmatrix} 2 & 7 \\ 3 & 11 \end{bmatrix}$$. 2. **Formula for inverse of a 2x2 matrix:** For $$A = \begin{bmatrix} a

1. **Problem Statement:** A linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ has eigenvalues (Băzel numbers) 1, 2, and 3.

1. **Problem statement:** Given a 3x3 matrix $A$ with characteristic polynomial $$P(\lambda) = (\lambda - 3)(\lambda + 1)^2,$$ answer the following: a) List all eigenvalues with th

1. **Problem Statement:** Given a 3x3 matrix $A$ with characteristic polynomial $$P(\lambda) = (\lambda - 3)(\lambda + 1)^2,$$ we need to:

1. **Problem Statement:** We have a linear transformation $T$ on the space of polynomials $P_2$ defined by $T(f(x)) = f'(x)$, where $f'(x)$ is the derivative of $f(x)$. We need to

1. **Problem Statement:** Given the linear transformation $T(x,y,z) = (2x - y, x + 4y, 3z)$, find: A) The matrix of $T$ relative to the standard basis of $\mathbb{R}^3$.

1. **Problem Statement:** Determine the geometric multiplicity and algebraic multiplicity of the eigenvalue $5$ for the matrix $$A = \begin{bmatrix}3 & 2 & 0 \\ 0 & 1 & 3 \\ -4 & -

1. **Find eigenvalues, eigenvectors, and basis of eigenspaces for matrix** (i) Given $$A=\begin{bmatrix}2 & 0 & 0 \\ 0 & 4 & 5 \\ 0 & 4 & 3\end{bmatrix}$$

1. The problem is to find the inverse of the matrix $$A = \begin{pmatrix} -1 & -3 & -1 \\ 0 & 0 & -2 \\ -1 & 1 & 1 \end{pmatrix}$$. 2. To find the inverse of a 3x3 matrix, we use t

1. **Problem Statement:** We want to find the smallest eigenvalue and its corresponding eigenvector of the matrix $$A = \begin{bmatrix} 5 & 2 & -1 \\ 2 & 1 & 1 \\ -3 & 3 & 4 \end{b

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. **Problem Statement:** Given polynomials $p_0(x), p_1(x), p_2(x), \ldots, p_n(x)$ where each $p_k(x)$ is a polynomial of degree $k$ for $k=0,1,2,\ldots,n$, prove that the set $\

1. **Problem Statement:** We are given matrices $A$, $B$, $C$, $D$ and vectors $u$, $v$, $w$, $E$. We need to evaluate expressions involving addition and scalar multiplication of t

1. You mentioned you have a linear algebra exam tomorrow. 2. To prepare effectively, focus on key topics such as vector spaces, matrix operations, determinants, eigenvalues, and li

1. The problem involves addition and scalar multiplication of matrices and vectors. 2. The general rules are:

1. **State the problem:** Solve the system of linear equations using Gaussian elimination: $$\begin{cases} x_1 + 3x_2 - x_3 = 2 \\ 2x_1 - x_2 + x_3 + 3x_4 = 14 \\ -3x_1 - 2x_2 + 4x

1. **Problem statement:** Calculate the matrix expression $3A + 4B$ where $$A = \begin{pmatrix}7 & -2 & 3 & -4 \\ 0 & 2 & 1 & -1 \\ -5 & 3 & 2 & 0\end{pmatrix}, \quad B = \begin{pm

1. **Problem 1(a): Find the rank of the matrix** $$\begin{bmatrix}1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 4 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 5 & 6\end{bmatrix}$$

1. **State the problem:** Find the rank of the matrix $$\begin{bmatrix} 1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 2 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 5 & 6 \end{bmatrix}$$

1. The problem is to perform the subtraction of two matrices: $$\begin{bmatrix}9 & -36 \\ 14 & 8 & 36 \\ 11 & 44 & 18\end{bmatrix} - \begin{bmatrix} \text{?} \end{bmatrix}$$